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STRUCTURAL DYNAMICS: RECENT ADVANCES 
Proceedings of the 6th International Conference 
Volume II 


Proceedings of the Sixth International Conference on Recent Advances in 
Structural Dynamics, held at the Institute of Sound and Vibration Research, 
University of Southampton, England, from 14th to 17th July, 1997, co-sponsored by 
the US Airforce European Office of Aerospace Research and Development and the 
Wright Laboratories, Wright Patterson Air Force Base. 


Edited by 

N,S. FERGUSON 

Institute of Sound and Vibration Research, 
University of Southampton, Southampton, UK. 

H.F. WOLFE 

Wright Laboratory, 

Wright Patterson Air Force Base, Ohio, USA. 
and 
C MEI 

Department of Aerospace Engineering, 

Old Dominion University, Norfolk, Virginia, USA. 


© The Institute of Sound and Vibration Research, University of Southampton, UK. 

ISBN no. 0-85432-6375 


19970814 055 




VV T.- , .. 


PREFACE 


The papers contained herein were presented at the Sixth International 
Conference on Recent Advances in Structural Dynamics held at the Institute of 
Sound and Vibration Research, University of Southampton, England in July 1997. 
The conference was organised and sponsored by the Institute of Sound and 
Vibration Research and co-sponsored by the Wright Laboratories, Wright Patterson 
Air Force Base. We wish to also thank the following for their contribution to the 
success of the conference: the United States Air Force European Office of Aerospace 
Research and Development. The conference follows equally successful conferences 
on the same topic held at Southampton in 1980,1984,1988,1991 and 1994. 

There are over one hundred papers written by authors from approximately 
20 different countries, making it a truly international forum. Many authors have 
attended more than one conference in the series whilst others attended for the first 
time. 


It is interesting to note the change in emphasis of the topics covered. 
Analytical and numerical methods have featured strongly in all the conferences. 
This time, system identification and power flow techniques are covered by even 
more papers than previously. Also, there are many contributions in the field of 
passive and active vibration control. Papers dealing with nonlinear aspects of 
vibration continue to increase. These observations seem to reflect the trend in 
current research in structural dynamics. We therefore hope that the present series 
of International Conferences will play a part in disseminating knowledge in this 
area. 


We would like to thank the authors, paper reviewers and session chairmen 
for the part they played in making it a successful conference. 

My personal thanks go to the following individuals who willingly and 
enthusiastically contributed to the organisation of the event: 

Dr. H.F. Wolfe Wright Laboratories, WPAFB, USA 

Dr. C. Mei Old Dominion University, USA 

Mrs. M.Z. Strickland ISVR, University of Southampton, UK 

Grateful thanks are also due to many other members of ISVR who contributed to 
the success of the event. 


N.S. Ferguson 


Sixth International Conference on 
Recent Advances in Structural Dynamics 

Volume II 


Contents 

Page No. 

INVITED PAPER 

R.D. BLEVINS 

On random vibration, probability and fatigue 881 

ACOUSTIC FATIGUE I 

58. J. LEE and K.R. WENTZ 

Strain power spectra of a thermally buckled plate in 

random vibration 903 

59. S.A. RIZZI and T.L. TURNER 

Enhanced capabilities of the NASA Langley thermal 

acoustic fatigue apparatus 919 

60. I. HOLEHOUSE 

Sonic fatigue characteristics of high temperature materials 

and structures for hypersonic flight vehicle applications 935 

61. M. FERMAN and H.F. WOLFE 

Scaling concepts in random acoustic fatigue 953 

ACOUSTIC FATIGUE II 

62. H.F. WOLFE and R.G. WHITE 

The development and evaluation of a new multimodal 

acoustic fatigue damage model 969 

63. B. BENCHEKCHOU and R.G. WHITE 

Acoustic fatigue and damping technology in composite 
materials 981 

64. D. MILLAR 

The behaviour of light weight honeycomb sandwich panels 

under acoustic loading 995 

65. P.D. GREEN and A. KILLEY 

Time domain dynamic Finite Element modelling in acoustic 

fatigue design 1007 



SYSTEM IDENTIFICATION II 


66. 

U. PRELLS, A.W. LEES, M.I. FRISWELL and M.G. SMART 
Robust subsystem estimation using ARMA-modelling in 
the frequency domain 

1027 

67. 

Y.Q. NI, J.M. KO and C.W. WONG 

Mathematical hysteresis models and their application to 
nonlinear isolation systems 

1043 

68. 

M.G. SMART, M.I. FRISWELL, A.W. LEES and U. PRELLS 

The identification of turbogenerator foundation models 
from run-down data 

1059 

69. 

C. OZTURK and A. BAHADIR 

Shell mode noise in reciprocating refrigeration 
compressors 

1073 

70. 

T.H.T. CHAN, S.S. LAW and T.H. YUNG 

A comparative study of moving force identification 

1083 

71. 

P.A. ATKINS and J.R. WRIGHT 

Estimating the behaviour of a nonlinear experimental multi 
degree of freedom system using a force appropriation 
approach 

1099 


POWER FLOW TECHNIQUES II 


72. 

R.S. LANGLEY, N.S. BARDELL and P.M. LOASBY 

The optimal design of near-periodic structures to minimise 
noise and vibration transmission 

1113 

73. 

J.L. HORNER 

Effects of geometric asymmetry on vibrational power 
transmission in frameworks 

1129 

74. 

M. IWANIEC and R. PANUSZKA 

The influence of the dissipation layer on energy flow in 
plate connections 

1143 

75. 

H. DU and F.F. YAP 

Variation analysis on coupling loss factor due to the third 
coupled subsystem in Statistical Energy Analysis 

1151 

76. 

S.J. WALSH and R.G. WHITE 

The effect of curvature upon vibrational power 
transmission in beams 

1163 

77. 

S. CHOI, M.P. CASTANIER and C. PIERRE 

A parameter-based statistical energy method for mid¬ 
frequency vibration transmission analysis 

1179 


PASSIVE AND ACTIVE CONTROL III 


78. Y. LEI and L. CHEN 

Research on control law of active suspension of seven 

degree of freedom vehicle model 1195 

79. M. AHMADIAN 

Designing heavy truck suspensions for reduced road 
damage 1203 

80. A.M. SADRI, J.R. WRIGHT and A.S. CHERRY 

Active vibration control of isotropic plates using 
piezoelectric actuators 1217 

81. S.M. KIM and M.J. BRENNAN 

Active control of sound transmission into a rectangular 
enclosure using both structural and acoustic actuators 1233 

82. T.J. SUTTON, M.E. JOHNSON and S.J. ELLIOTT 

A distributed actuator for the active control of sound 
transmission through a partition 1247 

83. J. RO, A. A-ALI and A. BAZ 

Control of sound radiation from a fluid-loaded plate using 

active constraining layer damping 1257 


ANALYTICAL METHODS 11 

84. E. MANOACH, G. DE PAZ, K. KOSTADINOV and 
F. MONTOYA 

Dynamic response of single-link flexible manipulators 1275 

85. B. KANG and C.A. TAN 

Wave reflection and transmission in an axially strained, 
rotating Timoshenko shaft 1291 

86. Y. YAMAN 

Analytical modelling of coupled vibrations of elastically 
supported channels 1329 

87. R.S. LANGLEY 

The response of two-dimensional periodic structures to 
harmonic and impulsive point loading 1345 

NONLINEAR VIBRATION III 

88. H. OYANG, J.E. MOTTERSHEAD, M.P. CARTMELL and 
M.L FRISWELL 

Stick-slip motion of an elastic slider system on a vibrating 
disc 


1359 


89. 

R.Y.Y. LEE, Y. SHI and C. MEI 

A Finite Element time domain multi-mode method for 
large amplitude free vibration of composite plates 

1375 

90. 

P. RIBEIRO and M. PETYT 

Nonlinear forced vibration of beams by the hierarchical 
Finite Element method 

1393 

91. 

K.M. HSIAO and W.Y. LIN 

Geometrically nonlinear dynamic analysis of 3-D beam 

1409 

92. 

R.Y.Y. LEE, Y. SHI and C. MEI 

Nonlinear response of composite plates to harmonic 
excitation using the Finite Element time domain modal 
method 

1423 

93. 

C.W.S. TO and B. WANG 

Geometrically nonlinear response analysis of laminated 
composite plates and shells 

1437 


ANALYTICAL METHODS III 


94. 

R.S. HWANG, C.H.J. FOX and S. McWILLIAM 

The free, in-plane vibration of circular rings with small 
thickness variations 

1457 

95. 

D.J. GORMAN 

Free vibration analysis of transverse-shear deformable 
rectangular plates resting on uniform lateral elastic edge 
support 

1471 

96. 

R.G. PARKER and C.D. MOTE, Jr. 

Wave equation eigensolutions on asymmetric domains 

1485 

97. 

A.V. PESTEREV 

Substructuring for symmetric systems 

1501 


RANDOM VIBRATION I 


98. 

G.FUandJ. PENG 

Anaytical approach for elastically supported cantilever 
beam subjected to modulated filtered white noise 

1517 

99. 

S.D. FASSOIS and K. DENOYER 

Linear multi-stage synthesis of random vibration signals 
from partial covariance information 

1533 

100. 

CW.S.TOand Z. CHEN 

First passage time of multi-degrees of freedom nonlinear 
systems under narrow-band non-stationary random 
excitations 

1549 


101. 

C. FLORIS and M.C. SANDRELLI 

Random response of Duffing oscillator excited by quadratic 
polynomial of filtered Gaussian noise 

1565 

102. 

S. McWILUAM 

Extreme response analysis of non-linear systems to random 
vibration 

1581 

103. 

M. GHANBARI and J.F. DUNNE 

On the use of Finite Element solutions of the FPK equation 
for non-linear stochastic oscillator response 

1597 


RANDOM VIBRATION II 


104. 

T.L. PAEZ, S. TUCKER and C. O’GORMAN 

Simulation of nonlinear random vibrations using artificial 
neural networks 

1613 

105. 

D.Z. LI and Z.C. FENG 

Dynamic properties of pseudoelastic shape memory alloys 

1629 

106. 

Z.W. ZHONG and C. MEI 

Investigation of the reduction in thermal deflection and 
random response of composite plates at elevated 
temperatures using shape memory alloys 

1641 


SIGNAL PROCESSING I 


107. 

M. FELDMAN and S. BRAUN 

Description of non-linear conservative SDOF systems 

1657 

108. 

N.E. KING and K. WORDEN 

A rational polynomial technique for calculating Hilbert 
transforms 

1669 

109. 

D.M. LOPES, J.K. FIAMMOND and P.R. WHITE 

Fractional Fourier transforms and their interpretation 

1685 


SYSTEM IDENTIFICATION III 

no. J. DICKEY, G. MAIDANIK and J.M. D’ARCHANGELO 

Wave localization effects in dynamic systems 1701 

111. P. YUAN, Z.F- WU and X.R. MA 

Estimated mass and stiffness matrices of shear building 
from modal test data 


1713 



112. YU. I. BOBROVNirSKn 

The problem of expanding the vibration field from the 
measurement surface to the body of an elastic structure 1719 

113 M. AMABILI and A. FREGOLENT 

Evaluation of the equivalent gear error by vibrations of a 
spur gear pair 


1733 




ON RANDOM VIBRATION, PROBABILITY, AND FATIGUE 


R. D. Blevins 

Rohr Inc., Mail Stop 107X 

850 Lagoon Drive 

Chula Vista, California 91910 


ABSTRACT 

Analysis is made to determine the properties of a random process consisting of the 
sum of a series of sine waves with deterministic amplitudes and independent, random 
phase angles. The probability density of the series and its peaks are found for an arbitrary 
number of terms. These probability distributions are non-Gaussian. The fatigue resulting 
from the random vibration is found as a function of the peak-to-rms ratio. 


1. INTRODUCTION 

Vibration spectra of aircraft components often are dominated by a relatively small 
number of nearly sinusoidal peaks as shown in Figure 1. The time history of this process, 
shown in Figure 2, is irregular but bounded. The probability density of the time history, 
shown in figure 3 only roughly approximates a Gaussian distribution and it does not exceed 
2.5 standeird deviations. 

The time history of displacement or stress of these processes over a flight or a take 
off time can be expressed as a Fourier series of a finite number of terms over the finite 
sampling period T. 

N 

y = 0,nCOs(u)-ntn + <^n), 0 < in < T, Un > 0 (l) 

n=l 

Each frequency Un is a positive, non-zero integer multiple of 27r/r. The following model is 
used for the nature of the Fourier series: 1) the amplitudes a-n are positive and deterministic 
in the sense that they do not vary much from sample to sample, 2) the phases (j>n are random 
in the sense that they vary from sample to sample, they are equally likely to occur over 
the range -oo < 0n < oo. This last condition implies that the terms on the right hand 
side of equation (1) are statistically independent of each other. 

We can generate an ensemble of values of the dependent variable Y by randomly 
choosing M sets of N phase angles = 1,2..^), computing Y at some ffxed time 

from equation (1), choosing another set of phases, computing a second value of Y and so 


881 


on until we have a statistically significant sample of M Y's. This random phase approach, 
introduced by Rayleigh (1880), models a multi-frequency processes where each frequency 
component is independent and whose power spectral density (PSD) is known. 

The maximum possible (peak) value of equation (1) is the sum of the amplitude of 
each term (recall > 0). The mean square of the sum of independent sine waves is the 
sum of the mean squares of the terms. 


N 


^peak ~ 'y ^ 


n=l 


= Na, for ai= 02 = an = CL 


(2a) 

( 2 &) 


X N N 

Yrms = ^ f [Y anCO«(2wt„/T + <Pri)?dtn = 5 

= |iVa^, for ai = 02 = an = a 

The peak-to-rms ratio of the sum of N mutually independent sine waves thus is, 

N N 


I rms 


1/2 


n=l 71=1 

= (2A/')^/^, for Oi = 02 = an = o. 


(3a) 

{Sb) 

(4a) 

(46) 


Equation (4b) shows that the peak-to-rms ratio for an equal amplitude series increases from 
2^/^ for a single term (N=l) and approaches infinity as the number of terms N approaches 
infinity, as shown in Figure 4. The probability of Y is zero beyond the peak value. For 
example, there is no chance that the sum of any four {N = 4) independent sinusoidal terms 
will be greater than 8^/^ = 2.828 times the overall rms value. 


2. PROBABILITY DENSITY OF A SINE WAVE 

The probability density py (y) of the random variable Y is probability that the random 
variable Y has values within the small range between y and y -b dy, divided by dy. p(Y) 
has the units of 1/Y. Consider single a sine wave of amplitude a^, circular frequency uJri) 
and phcLse 

Y = On COs{0Jntn + 0 < < 277. (5) 

Y is the dependent random variable. The independent random variables are tn or <l>n- The 
probability density of a sine wave for equal likely phases p((l>n) = l/(27r), or equally likely 
times, p[tn) = 1/T, is (Bennett, 1944; Rice 1944, art. 3.10), 


wiy) = 


77 ^(a^ - j/2) 1/2^ if <y <an\ 


0 , 


if I 2 /I > CLn 


( 6 ) 


882 



The probability density of the sine wave is symmetric about y = 0, i.e., pyiv) ~ PYi~y)) 
it is singular at y = Cn, and it falls to zero for jyl greater than an as shown in Figure 5. 

The characteristic function of a random variable x is the expected value of 

C{f) = r (7) 

J —CO 

and it is also the Fourier transform of the probability density function (Cramer, 1970, 
pp. 24-35; Sveshnikov, 1965; with notation of Bendat, 1958). j = is the imaginary 
constant. The characteristic function of the sine wave is found using equations (13) and 
(14) and integrating over the range 0 < X < a^. (Gradshteyn, Ryzhik, Jeffrey, 1994, article 
3.753). 


Cn(f) = 2(7ra„)-‘ T" cos{27rfY))[l - {Y/dY = Jo(27r/a„), (8) 

Jo 

The characteristic function of a sine wave is a Bessel function of the first kind and zero 
order (Rice, 1944, art. 3.16). Equations (6) and (8) are starting points for determining 
the probability density of the Fourier series. 


3. PROBABILITY DENSITY OF THE SUM OF N SINE WAVES 

It is possible to generate an expression for the probability density of Fourier series 
(equation l) with 1,2,3,to any number of terms provided the sine wave terms are mutually 
independent. This is done with characteristic functions. The characteristic function of the 
sum of N mutually independent random variables (Y = Xi -j- X 2 + -■■i- Xj\/) is the product 
of their characteristic functions (Weiss, 1990, p.22; Sveshnikov, pp. 124-129), 

c(/) = r .. r e^2-/{^.+^=+- 

7 —00 7—00 

N ^00 N 

= n / = n CM)- (9) 

The symbol 11 denotes product of terms. The characteristic function for the sum of N 
independent sine waves is found from equations (8) and (9). 


C{f)^ 


n!Li *^ 0 (27r/an), unequal an 
[Jo( 27 r/a)]^, ai == a2 = = a 


( 10 ) 


The probability density of Y is the inverse Fourier transform of its characteristic function 
(Sveshnikov, 1968, p. 129). 


py{y) = r e-^^-fyc{f)df 

7—00 


( 11 ) 


883 



By substituting equation (10) into equation (11) we obtain an integral equation for the 
probability density of a N-term finite Fourier series of independent sine waves (Barakat, 
1974). 


.oo N 

Pviy) = 2 / cos{2Tryf) { TT Jo(27r/an) } df, 

''0 n=l 


iV = 1,2,3... 


( 12 ) 


If all N terms of the Fourier series have equal amplitudes a = ai — a 2 = On = a/sr, then 
this simplifies, 


py(y) = 2 r cos(2iryf)[Jo{2wfa)fdf, N = 1,2,3... (13) 

Jo 

These distributions are symmetric about y — 0 as are all zero mean, sum-of-sine-wave 
distributions. Figures 5 and 6 show results of numerically integrating equations (45) and 
(46) over interval / = 0 to / = 15a using Mathematica (Wolfram, 1995). 

Barakat (1974, also see Weiss, 1994, p. 25) found a Fourier series solution to equation 
(45). He expanded the probability density of the N term sum in a Foui'ier series over the 
finite interval -Ly < Y < Ly where Ly = ai + a 2 + - + The result for unequal 
amplitudes is , 


\v\<Ly. 

i=L n=l 

For equal amplitudes, ai = a 2 = a,^ = o, Ly = Na, and 


i=l 


\y\ < Na. 


(14) 


(15) 


Figure 6 shows that the Fourier series solution (equation 15) carried to 20 terms to be 
virtually identical to numerical integration of equation (13) and it compares well with the 
approximate solution. Note that theory requires py{\yT\ > Ly) = 0. 

A power series solution for equation (13) can be found with a technique used by Rice 
(1944, art. 16) for shot noise and by Cramer (1970) who called it an Edgeworth series. The 
Bessel function term in equation (13) is expressed as an exponent of a logarithm which is 
then expanded in a power series, 

[Jo(27r/a)]-^ = ex'p{N ln[Jo[2'Kaf)\), (16) 

= exvi-Nir'^a^f - (l/4)iV7r^o'‘/'' - (l/9)N-K^a^f + (n/192)Arx5a®/-)- 

= + -1 

Substituting this expansion into equation (13) and rearranging gives a series of integrals, 
which are then solved (Gradshteyn, Ryzhik, Jeffrey, 1994, arts. 3.896, 3.952) to give a 


884 


power series for the probability density of the equal-amplitude N-term Fourier series sum. 


VY{y) = 




g^iFi[-2,V2,yV(2yA.,)l 




-( 


11 


r(9/2) 


(17) 


192iV3 32iv2^ ^1/2 


)^^iFi[-4,l/2,yV(2i;L.)l + -)^ \y\<Na 


'PY{\y\ > = 0 and Yrms is given by equation (3b). There are two special func¬ 

tions in equation (17), the gamma function T and the confluent hypergeometric function 
iFi[n,'y,z]. These are defined by Gradshteyn, Ryzhik, and Jeffrey (1994). 


As N approaches infinity, the peak-to-rms (equation 4b) ratio approaches infinity, and 
equation (51) approaches the normal distribution, 

\im pY{y) = ~;^—. (18) 

N-*oo V^Yrms 

as predicted by the central limit theorem (Cramer, 1970; Lin, 1976). 


4. PROBABILITY DENSITY OF PEAKS 

Theories for calculating the fatigue damage from a time history process generally 
require knowledge of the peaks and troughs in the time history. This task is made simpler 
if we assume that the time history is narrow band. If Y{t) is narrow band that is, that 
each trajectory of Y{t) which crosses zero has only a single peak before crossing the cixis 
again, then (1) the number of peaks equals the number of times the time history crosses 
the axis with positive slope, and (2) only positive peaks occur for Y{t) >0 and they are 
located at points of zero slope, dY{t)ldt = 0. Lin (1967, p. 304) gives expressions for the 
expected number of zero crossings with positive slope (peaks above the axis) per unit time 
for a general, not necessarily narrow band, process, 

ElNo+]= f ypyy(0,y)<iy (19) 

Jo 

and the probability density of the peaks for a narrow band process. 

= (20) 

In order to apply these expressions, the joint probability distribution of Y and Y must be 
established. The joint probability density function Pyriy^y) random variable 

Y and Y is the probability that Y falls in the range between y and y + dy and y falls in the 
range between y and y-\-dy, divided by dydy. The derivative of the sine wave Y (equation 
12) with respect to time can be expressed in terms of Y, 

dY/dt = Y — -Gn^n sin{u)nt + 0 n) == ~ Y^, \Y\ < On. ( 21 ) 


885 


The joint probability density is the inverse Fourier transform of its characteristic 
function, 

/ CO poo 

/ (30) 

-oo J—oo 

The proof of equations (28), (29), and (30) can be found in Chandrasekhar (1943), Willie 
(1987), Weiss and Shmueli (1987), and Weiss (1994, pp. 21-26). 

Since the probability is symmetric about y — y = 0, Pyriv^v) ~ Pyy(~S/j“y)) only 
symmetric terms survive the integration. Substituting, equation (29) into equation (30) 
and expanding gives and integral expression for the joint probability of y and Y. 

Pyy(2/,y)== [ [ {JJ ^o(27ranY^/f + /|cj2)}cos(27r/iy)cos(27r/2y)d/i<i/2 (31) 


It is also possible to expand the joint probability of Y and Y in as double finite Fourier 
series. The result is: 


.. oo oo N I . 7 

Pyriv^y) = (^)2j}cos(i7ry/Ly)cos(/:7ry/L^) 


aik = 1, i, /c> 0; 1/2, i — Qork = 0; 1/4, k = 0 


(32) 

(33) 


The expected number of peaks per unit time and the probability distribution of the narrow 
band peaks is obtained by substituting this equation into equations (19) and (20) and 
integrating. The results are: 


n—1 


^0 k=0 


TTUr 




Ly ‘ 




(34) 


y OO OO iV 


:/-)' +(^P)}sin(i^A/Ly) 
JuY J^Y 


where 


( 1/8, i = /c = 0, 

_ J 1/4, i > 0, A: = 0 

- 1 (l/2)[(-l)'‘- - ll/(fc 7 r)^ i = 0,fc > 0, 

I [(-1)* - ll/(fe7r)2, i>0.fc>0. 


(35) 


(36) 


If the frequencies are closely spaced so uJn^<^ and hence Ly » uLy , then one positive 
peak is expected once per cycle, 


£;[iVo+]=a;/(27r) 


(37) 


886 


and the probability density of narrow band peaks becomes, 


N 




t =0 fc =0 


71=1 




(38) 


Figure 7 shows probability density of narrow band peaks for N=2,3, and 4 equal amplitude 
(tti = 1) equal frequency series using equation (38). Each sum in equation (38) was carried 
to 40 terms. 

A power series solution for equation (20) can be found if all N terms in the series have 
equal amplitude and frequency. The result is 


Pa(A) = 




Y2 


1 1 
4iV 4ivy2 


A^ 


32Nr,i, 


•h... 


In the limit as N becomes infinite these equations become, 


(39) 


pYi-(y>y) = 


IttY Y 

Zr/i j rms*^ rms 


(40) 


Pa{A) = 


y2 

^ rms 


(41) 


Equation (40) is in agreement with an expression given by Crandall and Mark (1963, p. 
47) and equation (41)is the Rayleigh distribution. 


Equations (20), (35), (38), and (41) are conservative when applied to non-narrow band 
processes in the sense that any troughs above the axis (points with y > 0 and dYjdt = 0 
but d^Yldt^ > 0) are counted as peaks (Lin, 1967, p. 304; Powell, 1958; Broch, 1963). 


Equations (35), (38) and (41) can provide probability distributions for peaks of narrow 
band processes as a function of the number of sine waves from one to infinity and thus 
they model random processes with peak-to-rms ratios from 2^/^ to infinity. 


5. FATIGUE UNDER RANDOM LOADING 

Fatigue tests are most often made with constant-amplitude sinusoidal loading. The 
number of cycles to failure is plotted versus the stress that produced failure and the data 
is often fitted with an empirical expression. MIL-HDBK-5G (1994) uses the following 
empirical expression to fit fatigue data, 

log iVy = + B 2 log{S^ - S 4 ), = 5(1 - R)^K (42) 

Here Nf is the cycles to failure during sinusoidal loading that has maximum stress 5 per 
cycle. R is the ratio of maximum to minimum stress during a cycle. R = —1 is fully 


887 



reversed stress cycling. Bi though B 4 are fitted parameters. With a little work, we can 
put this expression in the form used by Crandall and Mark (1963, p. 113). 

JV = cSJ*- (43) 

where Sd = 5(1 - - B 4 , c = 10 -®', and b = - 82 - For cycling in a time history that 

has non constant amplitude, Miner-Palmgren proposed that the accumulated damage is 
the sum of the ratios of the number of cycles at each amplitude to the allowing number of 
cycles to failure at that amplitude (equations 42 and 43). 

D = ^«(Si)/lV^(S,) (44) 

i 

where n{Si) is the number of cycle accumulated at stress amplitude Si and Nj is the 
number of stress cycles at this amplitude which would cause failure. 

Following Miles(1954) and Crandall and Mark(1963), the expected fractional damage 
for a random stress cycling in system with dominant cycling at frequency f in time t/, is 

= (45) 

where Pa{S) is the probability density of a stress cycle having amplitude S and Nf{S) is 
the number of allowable cycles to failure at this stress. Failure under random loading is 
expected when the expected damage is unity. Setting = 1 at time such that 

ftd = Nd, the inverse of the expected number of random vibration cycles to failure is 

This expression can be used to create a fatigue curve for random cycling given the proba¬ 
bility density of the random stress cycle amplitudes (p^(5)) and a fatigue curve (equation 
42 with parameters Bi though B 4 and R) for sinusoidal cycling. 

Substituting the probability density expression for narrow band amplitude (equation 
35) and for the fatigue curve(equations 42 or 43) into equation (46) and integrating, we 
obtain an expression for the expected number of cycles to failure as a function of the 
number of sine waves and their amplitudes. For N equal amplitude sine waves this is, 


<"->■■ - I ^1" 


(47) 


V (2 + 6)L,(l-B)®= ' 

+ 


iirBi 


(1 - R)^Ly 
mBi 




■ 2 ’ 

.1 3 


1 T - irr -^ f -** ^ 

(1 - 


iV(B^(l-B)®’-B4)2, 
ALlil - B)2B3 ' 

i^7r^(£„(l - B)®° - Bif . 
4L|(1 - H)2®» 


888 


Recall that for this case Ly = Na, the rms value is Y^ms — {l/2)Na and the peak- 
to-rms ratio is Peak/Yrms = y/2N (equations 2 though 4). is the generalized 

hypergeometric function which is a series of polynomials. It is described by Gradshteyn, 
Ryzhik, and Jeffrey (1994). 

It is also possible to establish the fatigue curve using the Rayleigh distribution (equa¬ 
tion 18) and the MIL-HDBK-5 fatigue curve (equation 42). The result is 


(2^'^r[l + 5]((1 - 


-BlF,[l + 


^ 3 Bl 

2’2’2y;2„,(i-ij)2B3« 


(48) 






Bl 


rr2Y,^^{i-RYB,^ 


_orf3 + il + 3 Bl 

4 2’ 2’ 2V;2„,(1 - ii)2B3 


iFi[..] is the confluent hypergeometric function which is described by Gradshteyn, Ryzhik, 
and Jeffrey (1994). 

Much of the complexity of these last two equations arises from the term B 4 which is 
associated with an endurance limit in the fatigue equation. That is, equation (42)predicts 
that sinusoidal stress cycling with stress less than 54 /( 1 —R )^=3 produces no fatigue damage. 
If we set B 4 = 0 to set the endurance limit to zero, then equation (48) simplifies to. 

This result for cycles to failure under Gaussian loading without an endurance limit is also 
given by Crandall and Mark (1963, p. 117). 

Equations (47), (48) and (49) allow us to compute the fatigue curves of a material 
under random loading from a fatigue curve generated under sinusoidal loading (equation 
43) for narrow band random processes with any peak-to-rms ratio from 2^/^ to infinity. 


6. APPLICATION 

Figure 8 is the MIL-HDBK-5G fatigue curve for aluminum 2024-T3 with a notch 
factor of Kt=4 under sinusoidal loading with various R values. The fitted curve shown in 
the figure, gives the following parameters for equation (42). 

Bl = 8.3, B 2 = -3.30, Bz = 0.66, B 4 = 8.4 


889 


The B 2 and B 3 are dimensionless. B 4 has the units of ksi, that is thousands of psi, and 
10^^ has units of These Bi,..B 4 are substituted into equations (43), (47), (48), 

and (49). 

The fatigue curves under random loading are computed as follows, 1) the number of 
sine waves N is chosen and this fixes the peak-to-rms ratio from equation (3b), 2) set of 
values of rms stresses are chosen and for eac h the corresponding sine waves amplitudes are 
computed using equation (3b), a = Srmsy/VN (note that the peak stress much exceed 
S4=8.5 ksi), and 3)the cycles to failure are calculated from equation (47) for finite peak- 
to-rms ratios and equation (48) for Gaussian loading (infinite peak-to-rms). 

For single sine wave, the peak-to-rms ratio is 2^/^, equation 4b, and the fatigue curve 
interms of rms stress is adapted from the empirical data fit (equations 42, 43) by substi¬ 
tuting 2 ^^‘^SrTns for the stress amplitude. 


Nd = c(2^/25.n..(l - - B^r^ (50) 

where b= -B 2 and c = 10-®^ Some results are shown in Figure 9 for R=-l. 

7. CONCLUSIONS 

Analysis has been made to determine the properties of a random process consisting of 
the sum of a series of sine waves with deterministic amplitudes and random phase angles. 
The joint probability density of the sum and its first two derivatives is determined. The 
probability density of the sum and narrow band peaks have been found for an arbitrary 
number of statistically independent sine wave terms. The fatigue cycles-to-failure resulting 
from these processes has been found. 

1. The peak-to-rms ratio of the sum of mutually independent terms exceeds unity. If ail 
terms have the same peak and rms values then the peak-to-rms ratio of the series sum 
increases with the square root of the number of terms in the series. The probability 
of the series sum is zero beyond a maximum value, equal to the sum of the series 
amplitudes, and below the minimum value. Hence, he probability densities of the 
finite series, their peaks, and their envelope are non Gaussian. 

3. The formulas allow the direct calculation of the probability density of the series and its 
peaks from its power spectra density (PSD) under the assumption that each spectral 
component is statistically independent. 

4. The fatigue curves of a material under random loading with any peak-to-rms ratio 
from 2^/^ to infinity can be computed dfrectly from the fatigue curve of the material 
under sinusoidal loading. 


890 


REFERENCES 


Abramowitz, M. and LA. Stegun 1964 Handbook of Mathematical Functions, National 
Bureau of Standards, U.S. Government Printing OfRce, Washington D.C. Reprinted by 
Dover. 

Bennett, W.R., 1944 Acoustical Society of America 15, 165. Response of a Linear Rectifier 
to Signal and Noise. 

Bendat, J.S., 1958 Principles and Applications of Random Noise Theory, Wiley, N.Y. 

Chandrasekhar, S., 1943,Reweiys of Modem Physics, 15, 2-74. Also available in Wax, N. 
(ed) Selected Papers on Noise and Stochastic Processes, Dover, N.Y., 1954. 

Cramer, H., 1970 Random Variables and Probability Distributions, Cambridge at the Uni¬ 
versity Press. 

Crandall, S.H., and C. H. Mark 1963 Random Vibrations in Mechanical Systems, Academic 
Press, N.Y. 

Department of Defense, 1994 Metallic Materials and Elements for Aerospace Vehicle Struc¬ 
tures, MIL-HDBK-5G. 

Gradshteyn, I.S., I.M. Ryzhik, and A. Jeffrey 1994 Table of Integrals, Series, and Products 
5th Ed., Academic Press, Boston. 

Lin, P.K., 1976 Probabilistic Theory of Structural Dynamics, Krieger, reprint of 1967 edi¬ 
tion with corrections. 

Mathematica, 1995 Ver 2.2, Wolfram Research, Champaign, Illiinois. 

Miles, J., 1954 Journal of Aeronautical Sciences 21, 753-762. On Structural Fatigue under 
Random Loading. 

Powell, A., 1958 Journal of the Acoustical Society of America SO No. 12, 1130-1135. On 
the Fatigue Failure of Structure due to Vibrations Excited by Random Pressure Fields. 

Rayleigh, J.W.S. 1880 Philosophical Magazine X 73-78. On the Resultant of a Large 
Number of Vibrations of the Same Pitch and Arbitrary Phase. Also see Theory of Sound, 
Vol 10, art. 42a, reprinted 1945 by Dover, N.Y.. and Scientific Papers, Dover, N.Y., 1964, 
Vol. I, pp. 491-496. 

Rice, S.O., 1944 The Bell System Technical Journal 23 282-332. Continued in 1945 24 , 
46-156. Mathematical Analysis of Random Noise. Also available in Wax, N. (ed) Selected 
Papers on Noise and Stochastic Processes, Dover, N.Y., 1954. 

Shmulei, U. and G.H. Weiss 1990 Journal of the American Statistical Association 85 6-19. 
Probabilistic Methods in Crystal Structure Analysis. 

Sveshnikov, A.A, 1968 Problems in Probability Theory, Mathematical Statistics and Theory 
of Random Functions Dover, N.Y., translation of 1965 edition, pp. 74, 116. 


891 



Tolstov, G.P., 1962 Fourier Series, Dover, N.Y., pp. 173-177. Reprint of 1962 edition. 

Weiss, G.H., 1994 Aspects and Applications of the Random Walk, North-Holland, Amster¬ 
dam. 

Weiss, S.H. and U. Shmulei, 1987 Physica 146A 641-649. Joint Densities for Random 
Walks in the Plane. 

Willie, L.T., 1987 Physica 141A 509-523. Joint Distribution Function for position and 
Rotation angle in Plane Random Walks. 

Wirsching, RH., T.L. Paez, and K. Ortiz 1995 Random Vibrations, Theory and Practice, 
Wiley-Interscience, N.Y., pp. 162-166. 


892 


A 

an 

Bn 

C{f) 

cih.h) 

E[N,] 

E[No^] 

iFi 

pEq 

f 

i 

3 

Jo 

k 

K 

Ly 

Ey 

m 

N 

Nf 

n 

Bviy) 

Py(^) 

pxYi^,y) 

S 

t 

T 

Y 

Y 
X 

aij 

r 

lij 

(f>n 


U 

a>n 


NOMENCLATURE 

amplitude, peak, or envelope 

amplitude of the nth sine wave, a^, > 0 

fitted parameter in equation (42) 

characteristic function with parameter / 

joint characteristic function with parameters fi and /2 

expected number of positive peaks per unit time 

expected number of zero crossing with positive slope per unit time 

confluent hypergeometric function (Gradshteyn, Ryzhik, Jeffrey, 1994, art. 9.210) 

generalized hypergeometric function (Gradshteyn, Ryzhik, Jeffrey, 1994, art. 

9.210) 

parameter in Fourier transform 
integer index 
imaginary constant, 

Bessel function of first kind and zero order 
integer index 

complete elliptic integral of first kind, equation (33a) 

+ ^2 + •• + sum of amplitudes 
ujiai + uj 2 a 2 + - • + sum of velocity amplitudes 

integer index 
number of terms in series 
cycles to failure 
integer index, n=l,2,..N 

cumulative probability, the integral of Py{x) from x=—co to y 
probability density of random parameter Y evaluated at T = a: 
joint probability density of X and Y evaluated at Y = y and X = x 
stress 

time, 0 < t < T 

length of time interval 

sum of N modes or terms, —Ly<Y < Ly 

first derivative with respect to time of Y, —Ly <Y< Ly 

a random variable 

dimensionless coefficient, equation (33) 

gamma function, r[(2n + l)/2] = 7r^/^2“’^(2n - 1)!! 

dimensionless coefficient, equation (36) 

Dirac delta function 
3.1415926.. 

XiX 2 .-xi\i, product of terms 

phase angle of the nth sine wave, a uniformly distributed independent random 
variable 

circular frequency, a positive (non zero) real number 

circular frequency of the nth term, a non zero integer multiple of 27r/T 


893 





Figure 1 Spectrum of vibration of a component on a turbojet engine cowling. Note the 
finite number of distinct peaks. 


894 





Figure 2 Sample of the time history associated with the spectrum of Figure 1. Note the 
signal is bounded, irregular and quasi sinusoidal. 


895 





- 4.0 0 0.0 2.0 

MO, OF S.D. 


Figure 3 Probability density of the time history of Figures 1 and 2. Note that the maximum 
values do not exceed plus or minus 2.5 standard deviations. 






Yrms * (Probability Density of Y) 



-3-2-10 1 2 3 

Y / Yrms 


- Normal Distribution 

-Sine Wave Distribution 

o Equation . N=10 
• Equation , N=1 


Figure 5 Normal probability density (equation 18) and sine wave probability density (equa¬ 
tion 6) in comparison with results of numerical integration of equation (13) for N=1 and 
N=10. 


898 






Yrms* (Probability Density of Y) 






Yrms (Probability Density of A) 



-Two Sine Waves 

-Three Sine Waves 

.Four Sine Waves 


Figure 7 Probability density of peaks in narrow band series with equal amplitudes (ai = 
02 .. = 1) and frequencies. 


900 





10 * 


in' {o' 10* 10* 10* 

FRTIGUE LIFE. CYCLES 

FIGURE 3.2.3.1.8(h). Besi-fit SI N curves for noiched. K, ~ 4.0 of 2024-T3 aluminum alloy sheet, 
longitudinal direction. 


Figure 8 Fatigue curves for notched 2024-T3 aluminum alloy with Kt=4. MI1-HDBK-5G 
(1994, p. 3-115) 
















902 


ACOUSTIC FATIGUE I 




Strain Power Spectra of a Thermally Buckled Plate 
in Random Vibration 

Jon Lee and Ken R. Wentz 
Wright Laboratory (FIB) 

Wright-Patterson AFB, OH 45433, USA 

Abstract 

Several years ago, Ng and Wentz reported strain power spectra measured 
at the mid-point of a buckled aluminum plate which is randomly excited by an 
electrodynamic shaker attached to the clamped-plate boundary fixture. We 
attempt to explain the peculiar features in strain power spectra by generating 
the corresponding power spectra by the numerical simulation of a single-mode 
equation of motion. This is possible because the essential dynamics takes place 
in the frequency range just around and below the primary resonance frequency. 

1. Introduction 

For high performance military aircraft and future high-speed civil transport 
planes, certain structural skin components are subjected to very large acoustic 
loads in an elevated thermal environment [1]. This is because high-speed 
flights call for a very powerful propulsion system and thereby engendering 
acoustic loads in the anticipated range of 135-175 dB. More importantly, 
because of the aerodynamic heating in hypersonic flights and the modern trend 
in integrating propulsion sub-systems into the overall vehicular configuration, 
some structural components must operate at high temperatures reaching up to 
1300°F. Hence, the dual effect of thermal and acoustic loading has given rise 
to the so-called thermal-acoustic structural fatigue [2,3]. 

Generally, raising the plate temperature uniformly but with an immovable 
edge boundary constraint would result in thermal buckling, just as one observes 
flexural buckling as the inplane stress along plate edges is increased beyond a 
certain critical value. This equivalence has been recognized [4,5] and 
exploited in previous analytical and experimental investigations of the thermal- 
acoustic structural fatigue [6,7,8]. An experimental facility for thermal- 
acoustic fatigue, termed the Thermal Acoustic Fatigue Apparatus, was 
constructed at the NASA Langley Research Center in the late 80’s. Under the 
acoustic loading of 140-160 dB, Ng and Clevenson [9] obtained some strain 
measurements of root-mean-square value and power spectral density (PSD) on 
an aluminum plate heated up to 250^. Later, Ng and Wentz [10] have 
repeated the heated Aluminum plate experiment but by randomly exciting the 
clamped-plate boundary fixture by a shaker, and thereby recovering similar 
strain measurements. 

It should be noted that Ng and his colleagues [7,9,10] were the first to 
achieve sufficient plate heating to induce thermal buckling and thus observe the 
erratic snap-through under the acoustic or shaker excitations. Here, by erratic 


903 


we mean that a snap-through from one static buckled position to another takes 
place in an unpredictable fashion. We reserve the adjective chaotic for a snap- 
through occurring under the deterministic single-frequency forcing [11,12]. It 
has already been observed that certain of the buckled plate experiment can be 
explained, at least qualitatively, by a single-mode model of plate equations. 
This is also validated by a theoretical analysis. Indeed, we showed that a single¬ 
mode Fokker-Planck formulation can predict the high-temperature moment 
behavior and displacement and strain histograms of thermally buckled plates, 
metallic and composite [13,14]. 

In retrospect, a single-mode model has proven more useful than originally 
intended. That is, the single-mode Fokker-Planck formulation of an isotropic 
plate lends itself to predicting certain statistics of composite plates which are 
simulated by multimode equations or tested experimentally by multimode 
excitations. For a refined and more quantitative comparison, one must inject 
more realism into dynamical models by including the multimode interactions. 
However, before giving up the single-mode plate equation, there is an 
important problem that this simple model is well suited for investigation. That 
is, prediction of the strain PSD measurement by Ng and Wentz [10]. As we 
shall see in Sec. 4, the strain PSD of a thermally buckled plate exhibits a strong 
spectral energy transfer toward zero frequency, and thereby saturating 
frequency range well below the primary resonance frequency. This downward 
spectral energy transfer can be modeled quite adequately by the single-mode 
plate equation without necessitating multimode interactions. 

2. Equation of motion for the aluminum plate experiment 

By the Galerkin procedure, the von Karman-Chu-Herrmann type of large- 
deflection plate equations give rise to infinitely coupled modal equations [15]. 
However, much has been learned from a prototype single-mode equation for 
displacement^ [13,14]. 

q + Pq + k„{l-s)q + aq^ = g„ + g{t), (1) 

where the overhead dot denotes d/dt and the viscous damping coefficient is 
P = 2^^ with damping ratio ^. For the clamped plate, we have 

;i„=f(r‘'+2rV3 + l), 

s=rji+(1 -M) (1+(r^+ ir^) /6], 
a = ^{(7^+r'^+2^i) + |(i-/i^)[T(r^+r'^) +^(.r+r''T^ 

+ (r+47'‘)‘^ + (47+ 7"‘r^]}, 

&= (r‘'+2r^/3 +i)Sjj6. 

Note that the expressions for s and g„ are specific to the typical temperature 


904 


variation and gradient profiles assumed in Ref. [15]. Here, 7 = b/a is the 
aspect ratio of plate sides a and b, and fi is Poisson's ratio. The uniform plate 
temperature is measured in units of the critical buckling temperature. The 
maximum temperature variation on the mid-plate plane is denoted by and 
TJ5g is the maximum magnitude of temperature gradient across the plate 
thickness, where 5^ and 5^ are scale factors. Hence, 0 signifies no 
temperature variation over the mid-plate plane, and 0 zero temperature 
gradient across the plate thickness. Finally, g{t) denotes the external forcing. 

The parameter s represents thermal expansion due to both the uniform 
plate temperature rise above room temperature and temperature variation over 
the mid-plate uniform temperature. The combined stiffness k^(l - s)q consists 
of the structural stiffness k^q and thermal stiffness -sk^q , which cancel each 
other due to the sign difference. It is positive for 5 <1, then Eq. (1) has the 
form of Duffing oscillator with a cubic term multiplied by a , which represents 
geometric nonlinearity of membrane stretching. For s >1 Eq. (1) reduces to 
the so-called buckled-beam equation of Holmes [11] with a negative combined 
stiffness. In contrast, denotes thermal moment induced by a temperature 
gradient across the plate thickness; hence, it appears in the right-hand side of 
Eq. (1) as an additional forcing. The interplay of the terms involving 5 , a, and 
g^ can best be illustrated by the potential energy [15] 

U{q) = -goq + k^(X-s)(fn -H a . (2) 

Fig. 1 shows that V{c^ is symmetric when g„ = 0. For s<l it has a single well 
which splits into a double well as s exceeds unity. Note that the distance 
between the twin wells increases as for large s (Fig. 1(b)). This 
interpretation is valid approximately for go>^- That is, a positive g^ lowers 
the positive side potential (^> 0 ) and raises the negative side potential {q<0), 
and thereby rendering the potential energy asymmetric. 

U(q) ^(^1) 




Fig. 1 Potential energy, (a) s<V, (b) ^ >1, w = ^k^(s - l)/a , d = - l))V4a. 

(- ^.= 0 ; --- 5 „> 0 ) 


905 



It must be pointed out that Eq. (1) is dimensionless and involves explicitly 
only 7 and /i. For the aluminum plate experiment [10], 7=10 in./8 in. and 
so that k^=Q3.9l and a=85.33. If we further assume 5^- 0 for 

simplicity, the thermal parameter reduces to s=T^. Previously, Eq. (1) was 
used for the investigation of stationary Fokker-Planck distribution which 
involves only the ratio p!F, where F is the constant power input [13,14]. 
Hence, nondimensionalization has indeed spared us from specifying in detail 
other plate parameters. Things are however different in numerical simulation 
because we must know the characteristic scales to correctly interpret time- 
dependent solutions. By retracing the derivation, we find that the dimensionless 
quantities in Eq. (1) are (Eq. (IV. 1) in Ref. [1]) 

q-qlh, t-t/t*, g = glg*, (3) 

where the overhead bar denotes the physical quantity. Here, the plate thickness 
h, t*={b/Kf.y[ph/D, and g^=p(h/t*)^ are the characteristic length, time, and 
force, respectively (p = mass density, D=Eh^f\2{\~p}), £= Young’s 
modulus of elasticity), as listed in Table L We now rewrite Eq. (1) with the 
numerical coefficients (Table I). _ 

q + 0.0978? + 23.91(1 -s)q + 85.33?^ = (4) 


where g{t) has the unit of psi. 

Table I. Parameter values for the aluminum plate experiment 


7, « 

10in./8in., VoX 0.01, 23.910, 85.332 


5.. 5, 

p 

o 


h, f*, g'^- 

0.05 in., 3.305 lO'^sec.^, 5.806 1 O'’psi^ 



(+) p = 0.098 Ib/in^ and E =1.03 10’ psi. 


3. Monte-Carlo simulation 

Because of 5 =0, Eq. (4) has the standard form of Duffing (s<l) and 
Holmes (s>l) oscillators. In stead of a single frequency for forcing g{t) [11, 
12], in Monte-Carlo simulation all forcing frequencies are introduced up to a 
preassigned maximum so that forcing represents a plausible physical 
realization. Of course, particular interest here is a constant PSD. We shall 
begin with generation of a time-series for random processes with such a PSD. 

3.1 Random forcing time-series 

We adopt here the procedure for generating a time-series of Shinozuka 
and Jan [16], which has been used for a oscillator study [17] and extensively 
for structural simulation applications by Vaicaitis [18,19]. Since it relies 
heavily on the discrete fast Fourier transforms, such as FFTCF and FFTCB 
subroutines of the IMSL library, it is more expedient to describe the procedure 
operationally rather than by presenting somewhat terse formulas. Let us 


906 


introduce Nj: frequency coordinates which are equally spaced in 
the band width A/=/^ 3 x/A^^. Now, the task is to generate a time-series of total 
time T that can resolve up to . Assume T is also divided into time 
coordinates with the equal time interval At=T/Nj, From the time-frequency 
relation r=l/A/, we find ^At. If we choose 

N 

Nf = ( 5 ) 

is the Nyquist frequency, consistent with our original definition of the 
upper frequency limit of resolution. 

A random time-series with a constant PSD can be generated in the 
following roundabout way. We begin by assuming that we already have a 
forcing power spectrum ^g{f) of constant magnitude over [0,/njax]- Such a 

PSD may be represented by a complex array A„= VC exp(~27rz0„) (n = 1, 
Nf), where takes a random value distributed uniformly in [0, 1]. Clearly the 
magnitudes of are C, hence We then enlarge the complex array 

A„ by padding with zeros for n =Ny+l, and Fourier transform it to 

obtain a complex array B^{n= N^). The random time-series for is 

now given by the real part of 

= Real part of (n = 1, N^) (6) 


As it turns out, when g „ is padded with zeros for the imaginary components 
and Fourier transformed, we recover the original array A„ (w=l, Nf) with 


Since the spectrum area is nothing but total forcing power 

<g^> (say, in psi^), we can relate C with the variance <g^> of pressure 
fluctuations, which is often expressed by the sound pressure level (SPL) in dB, 
according to SPL=10 log<g2 >/p2, where p=2,9 10*^ psi. Hence, 

c = -. (7) 

/max 

Here, Eqs. (6) and (7) defined heuristically are meant to explain the 
corresponding formulas (2) and (12) in Ref [18]. 

For the numerical simulation we first note that the resonance frequency of 
Eq. (1) is /^=-y^/27r« 0.778 for s ~ 0. This gives the dimensional resonance 

frequency fjt*~235.5 Hz which is somewhat larger than the experimental 
217.7 Hz (Fig. 3(a)). As shown in Table II, we assign (~9/r) because 

the electrodynamic shaker used in the experiment [10] has the upper frequency 
limit 2000 Hz. 


Table II. Dimensionless parameter values for the numerical simulation 

at, N, _ 7, 8192, 4096 

At, T 0.071. 585 


907 








3.2 Displacement power spectrum 

Under a random time integration of Eq. (4) yields a time-series for q^. 
We first comment on the time integration. Although there are special solvers 
[17,20] proposed for stochastic ordinary differential equations (ODEs), we 
shall use here the Adams-Bashforth-Moulton scheme of Shampine and Gordon 
[21], which has been implemented in DEABM subroutine of the SLATEK 
library. Although DEABM has been developed for nonstochastic ODEs, its 
use for the present stochastic problem may be justified in part by that one 
recovers linearized frequency response functions by the numerical simulation 
(Sec. 3.4). Obviously, this does not say anything about the strongly nonlinear 
problem in hand, and it should be addressed as a separate issue. In any event, 
DEABM requires the absolute and relative error tolerances, both of which are 
set at no larger than 10“^ under the single-precision algorithm for time 
integration. Note that actual integration time steps are chosen by the 
subroutine itself, commensurate with the error tolerances requested. Recall 
that is updated at every time interval Ar, and we linearly interpolate the 
forcing value within A?. 

We begin time integration of Eq. (4) from the initial configuration at the 
bottom of the single-well potential, ^(0)=p(0)=0, for 5' <1 and the positive 
side double-well potential, q{0)=^kj,s~\)/a andp(0)=0, for j >1. And we 
continue the integration up to T. By Fourier transforming time-series q^, we 
obtain displacement power spectrum 0^(/). This process of integrating and 
transforming is repeated over three contiguous time ranges of 7, and the 
successive PSDs are compared for stationarity. Since it is roughly stationary 
after three repetitions, we report here only the PSD of the third repetition. 
From the stationary input-output relation [22] where 

is the magnitude of system frequency response function, we write 

= (S) 

Since O (/)=C, the and would have a similar functional 

s 

dependence upon /, Hence, we call them both the displacement PSD. 

3.3 Strain power spectrum 

Although displacement is the direct output of numerical simulation, one 
measures strain rather than the displacement in plate experiment. At the 
present level of plate equation formulation, the strain e is given 

by the quadratic relation 

e = + C^q + C2(f‘ ■> (9) 

where C,- are given at the middle {x/a = y/b =1/2) of a clamped plate as follows 
(Appendix D of Ref [13]) 


908 


c = 


(/+2yV3+l)?;g, 

3(1+At)(y"+1) 



{l-li}- 



q = 


8 y^ 

3 ’ 


r 32 fy^ 5n (l-/iyV4) 

^ 9 [2 16 2(y+y-‘f (y+4y-‘)^ (4y + y‘fJ 


For we have C^=0, Q =4.17, and Q =2.77 (Table I). Hence, Eq. (9) 
engenders only the linear and quadratic transformations, but no translation. In 
any event, translation has no effect on the spectral energy contents. By Fourier 
transforming time-series (n=l, A^^), we obtain strain power spectrum 

. Although the forcing PSD is not constant, one computes the forcing 
spectrum ratio as in Eq. (8) and call it the magnitude square of strain frequency 
response function for the lack of a better terminology. 

3.4 The linear oscillators 

For the pre-buckled (5 <1) linear oscillator (a= 0) we rewrite Eq. (4) in 
standard form _ 

q + + 0)1(1 -s)q = (10) 

where col=k^, and obtain 

I H^(f)^ = [(0)1(1 -s)- + (An^co^ffT'- (11) 

As shown in Fig. 2(a), the numerical simulation of Eq. (10) recovers 
as given by Eq. (11) over the entire frequency range. Although the simulation 
of Fig. 2(a) was carried out with SPL=130 dB, it does not depend on SPL 
since Eq. (10) is linear. Physically speaking, Eq. (10) oscillates in a single-well 
potential (Fig. 1(a)). Since the potential energy has two wells (Fig. 1(b)) for 
s >1, we linearize Eq . (1) arou nd the positive side potential well by the 
transformation q=q'+^k^{s-l)la . Hence, the corresponding linear oscillator 

is 


(a) (b) 



Fig. 2 Linear frequency response functions, (a) Displacement; (b) Strain 
(j = 0; -Numerical simulation; • Eq. (U)) 


909 



q' + 2^0)^ q’ + Icolis -V)q' = (12) 

s 

In parallel to Eq. (11), the frequency response function of a post-buckled plate 

I = [(^-colis - 1) - 4;rV^)2 + . (13) 

The resonance frequency f=co^^2{s-l)/27J: of a post-buckled (s >1) plate 
should be compared with f=co^^2{\~s)/2n of the pre-buckled (j <1) plate. 

Now, for the linear oscillators we see that is also given by Eq. (11) 

and (13) for ^ <1 and >1, respectively (Fig. 2(b)). This is because the spectral 
energy distribution is not at ail affected by a linear transformation. 

4. Displacement and strain power spectra 

As we shall see in Sec 4.1, the experimental strain PSD exhibits downward 
spectral energy transfer toward zero frequency, so that there is a considerable 
spectral energy buildup below the resonance frequence as SPL is raised. 
Moreover, it also involves an upward spectral energy transfer which then 
contributes to both the increased resonance frequency and broadened 
resonance frequency peak. Since spectral energy transfers take place around 
and below the primary resonance frequency, it is possible to depict the 
downward and upward spectral energy transfers by the numerical simulation of 
Eq. (4) without necessitating multimode interactions. We shall first discuss the 
characteristic features of experimental strain PSDs. 

4.1 Experimental strain PSD 

Of the spectra reported in Ref. [10], we consider the following two sets. 
One is the nonthermal set (^=0) consisting of two PSDs of small and large 
SPLs. The other is the post-buckled set (5=1.7) of four PSDs. For the 
convenience of readers, we have reproduced in Figs. 3 and 4 the selected PSDs 
from Ref. [10] by limiting the upper frequency to 600 Hz, and the pertinent 
data are summarized in Table m. 


Table in. Strain power spectra of experiment and numerical simulation 



Fig. 4(a) Fig. 6a 


Fig. 4(b) Fig. 6b 


Fig. 4(c) Fig. 6c 
Fig. 4(d) Fig. 6d (*) 



Fig. 7 
Fig. 8 
Fig. 9 


Fig. 10 


(*) Computed from the acceleration a measured in units of g. 


910 









Fig. 4 Experimental strain PSD =1.7). (a) 130.1dB; (b) 142dB; (c) 151.5dB; (d) 154.6dB 

The following observations are drawn from the experimental PSDs. First, 
for the nonthermal plate 

Figure 3fa) : Compare the measured strain fr-2\l Hz with the theoretical 
displacement235 Hz of Eq. (4). Note that a small spectral energy peak is 
found at 467 Hz which is about twice (-2.15) the strain value. 


911 



Figure Sfb) : With SPL~150 dB the strain increases to 240 Hz and the 
spectral width at the half resonance peak has nearly doubled. The spectral 
energy buildups at zero and 515 Hz are more noticeable than in Fig. 3(a). 
Again, 515 Hz is about twice (-2.15) the primary strain f,. 


(a) (b) 



f f 


Fig. 5 Numerical simulation results under .y=0 and SPL=130 dB. 

(a) Displacement (-simulation,* Eq. (11)); (b) Strain (-simulation, • Eq. (11)); 

(c) PSD averaged over 12 frequency intervals (-displacement, —• — strain); 

(d) Strain PSD. 


Next, for the thermally buckled plate 

Figure 4(a) : The primary strain fr=227 Hz should be compared with the 
theoretical displacement /^=279 Hz of Eq. (13). A second spectral energy 
peak is found at 537 Hz, much larger than twice (-2.37) the primary strain /^. 
Figure 4rb) : Here, the spectral energy buildup is most significant at zero 
frequency. Besides, there appear two spectral energy humps at 100 and 183 
Hz, below the primary strain = 227 Hz of Fig. 4(a). Discounting the zero- 
frequency spectral peak, PSD may be approximated by a straight line in the 
semi-log plot, hence it is of an exponential form up to 400 Hz. 








Figure 4rc) : The zero-frequency peak is followed by a single spectral energy 
hump at 115 Hz. Again, PSD can be approximated by a straight line and its 
slope is roughly the same as in Fig. 4(b). 

Figure 4rd) : A major spectral energy peak emerges at 130 Hz, followed by a 
minor one at 350 Hz. Theoverall spectral energy level is raised so that the 
magnitude of PSD ranges over only two decades in the figure. 

In Figs. 4(b)-(d) we have ignored the spectral energy peaks at around 500 
Hz, for they are not related to the first plate mode under consideration. This is 
further supported by the simulation evidence to be discussed presently. 


4.2 Numerical simulation results 

After choosing .y = 0 or 1.7, we 
are left with SPL yet to be specified. o 
Ideally, one would like to carry out the 
numerical simulation of Eq. (4) by 
using SPL of the plate experiment 'm-z 
(Table III) and thus generate strain ^ 

PSDs which are in agreement with 
Figs. 3 and 4. Not surprisingly, the _4 
reality is less than ideal. An obvious 
reason that this cannot be done is that 
the forcing energy input is fed into all f 

plate modes being excited in Hg. 6 PSD averaged over 12 frequency 
experiment, whereas the forcing (j=0, SPL=138dB) 

energy excites only one mode in the - displacement; -•-strain 

numerical simulation. Consequently, SPL for the numerical simulation should 
be less than the experimental SPL, but we do not know a priori how much 
less. We therefore choose a SPL to bring about qualitative agreements 
between the single-mode simulation and multimode experiment. As anticipated, 
the simulation SPLs (Table HI) are consistently smaller than the experimental 
values. 

The numerical simulation results are shown in Figs. 5-6 for 5 = 0 and Figs. 
7-10 for s =1.7. Actually each figure has four frames, denoted by (a)-(d). 
First, frames (a) and (b) depict and Since they are very 

jagged at large SPLs, we average the spectral energy over 12 frequency 
intervals and present both of the smoothed-out frequency response functions in 
the same frame (c). Lastly, frame (d) shows Og(/) itself Since there is no 
qualitative difference between <E>g(/) and we shall call them both the 

strain PSD. We present all four frames (a)—(d) of Figs. 5 and 7, but only the 
frame (c) of Figs. 6, 8, 9 and 10 here for the lack of space. 

First, for the nonthermal plate 

Figure 5 : The simulated is closely approximated by Eq. (11) with f = 

236 Hz. Note that is also approximated by Eq. (11) for all frequencies 



913 



Fig. 7 Numerical simulation results under j=1.7 and SPL=129 dB. 

(a) Displacement (-simulation,* Eq. (13)); (b) Strain (-simulation, • Eq. (13)); 

(c) PSD averaged over 12 frequency intervals (-displacement, —• strain); 

(d) Strain PSD. 


but zero and 476 Hz, where the strain spectral energy piles up due to the 
quadratic transformation (9). Since 476 Hz is nearly twice (-2.02) the primary 
/^, strain spectral energy buildups are due to the sum and difference of two 

nearly equal frequencies, ± / 2 , where/i==/ 2 ^/^. 

Figure 6 : The primary strain is shifted slightly upward to 253 Hz and the 

spectral width at half resonance peak is 50% wider than that of Fig. 5(c). The 
spectral energy builds up at 525 Hz which is roughly twice (-2.08) the /^. At 
SPL=138 dB we find that the strain spectral energy hump at 525 Hz is about 2 
decades below the resonance frequency peak, as was in Fig. 3(b). 

Now, for the thermally buckled plate 

Figure 7 : The simulated and are weU approximated by Eq. 

(13) around /^=270 Hz which is a litde below the linearized /^=279 Hz. 
Unlike in Fig. 5 for 5=0, both and l/7^(/)F show spectral energy 

building up significantly near zero and 543 Hz which is twice (-2.01) the /^. 


914 




2p —!- 1 -!- 1 - 1 -T—n- 1 -r 



^_1-1-1-<-1->-1-^-1 

0 300 600 


f 

Fig. 8 PSD averaged over 12 frequency 
intervals (j =1.7, SPL=138 dB) 

-displacement; —• — strain 


Note that in Fig. 7(a) the spectral 
energy hump at 543 Hz is about 3 
decades below the primary frequency 
peak, as was in Fig. 4(a). 

Figure 8 : After a large zero-frequency 
peak, two spectral energy humps 
appear at 131 Hz and 236 Hz. Note 
that the ratios of these frequencies to 
the /, (131/279 -0.47 and 236/279 - 
0.85) are comparable with the same 
ratios (100/227 -0.44 and 183/227 - 
0.81) found in Fig. 4(b). Excluding 
the zero-frequency peak, the overall 
strain PSD is a straight line, hence of 
an exponential form, as in Fig. 4(b), 


Figure 9 : The zero-frequency spectral peak is followed by a single major 
energy hump at 154 Hz. The ratio of this to the (154/279 -0.56) is 
somewhat larger than the ratio (115/227 -0.51) in Fig. 4(c). The strain PSD 
can also be approximated by a straight line over the entire frequency range, and 
Figs. 8 and 9 seem to have the same slope when fitted by straight lines. 

Figure 10 : The spectral magnitude of is larger than that of in 

the frequency range above 300 Hz. The choice of SPL=146 dB was based on 
that the PSD magnitude around 300 Hz is about 2 decades below the main 
spectral peak magnitude at 180 Hz, thus emulating Fig. 4(d). 

All in all, by numerical simulations we have successfully reproduced the 
peculiar features in the two sets of strain PSDs observed experimentally under 5 
= 0 and 1.7. 



Fig. 9 PSD averaged over 12 frequency 
intervals (s =1.7, SPL=143 dB) 

-displacement; —•—strain 



Fig. 10 PSD averaged over 12 frequency 
intervals (j =1.7, SPL=146 dB) 

-displacement; —•—strain 


915 




5. Concluding remarks 

At low SPL the nonthermal {s= 0) and post-buckled (^=1.7) plates appear 
to have a similar PSD. However, this appearance is quite deceptive in that the 
nonthermal plate motion is in a single-well potential, so that PSD does not 
change qualitatively as SPL is raised. On the other hand, the trajectory of a 
post-buckled plate is in one of the two potential energy wells when SPL is very 
small. However, as we raise SPL such a plate motion can no longer be 
contained in a potential well, and hence it encircles either one or both of the 
potential wells in an erratic manner. This is why the experimentally observed 
and numerically simulated strain PSDs of a post-buckled plate exhibit 
qualitative changes with the increasing SPL, and thereby reflect the erratic 
snap-through plate motion. A quantitative analysis of snap-through dynamics 
will be presented elsewhere. 

Lastly, we wish to point out that a PSD of straigh-line form in the semi-log 
plot was observed in a Holmes oscillator when trajectories are superposed 
randomly near the figure-eight separatrix [23]. 

Acknowledgments 

Correspondence and conversations with Chung Fi Ng, Chuh Mei, Rimas 
Vaicaitis, and Jay Robinson are sincerely appreciated. We also wish to thank 
the referees for their helpful suggestions to improve the readability of this 
paper. 


References 

1. Lee, J., Large-Amplitude Plate Vibration in an Elevated Thermal 
Environment, WL-TR-92-3049, Wright Lab., Wright-Patterson AFB, OH, 
June, 1992. 

2. Jacobson, M.J. and Maurer, O.F., Oil canning of metallic panels in 
thermal-acoustic environment, AIAA Paper 74-982, Aug., 1974. 

3. Jacobson, M.J., Sonic fatigue of advanced composite panels in thermal 
environments, J. Aircraft, 1983, 20, 282-288. 

4. Bisplinghoff, R.L. and Pian, T.H.H., On the vibrations of thermally 
buckled bars and plates, in Proc. 9th Inter. Congr. of Appl. Mech., 
Brussels, 1957, 7, 307-318. 

5. Tseng, W.-Y., Nonlinear vibration of straight and buckled beams under 
harmonic excitation, AFOSR 69-2157TR, Air Force Office of Scientific 
Research, Arlington, VA, Nov., 1969. 

6. Seide, P. and Adami, C., Dynamic stability of beams in a combined 
thermal-acoustic environment, AFWAL-TR-83-3072, Flight Dynamics 
Lab., Wright-Patterson AFB, OH, Oct., 1983. 

7. Ng, C.F., Nonlinear and snap-through responses of curved panels to 
intense acoustic excitation, /. Aircraft, 1989, 26, 281-288. 

8. Robinson, J.H. and Brown, S.A., Chaotic structural acoustic response of a 
milled aluminum panel, 36th Structures, Structural Dynamics, and 


916 



Material Conference, AIAA-95-1301-CP, New Orleans, LA, 1240-1250, 
Apr. 10-13, 1995. 

9. Ng, C.F. and Clevenson, S. A., High-intensity acoustic tests of a thermally 
stressed plate, J, Aircraft, 1991, 28, 275-281.. 

10. Ng, C.F. and Wentz, K.R., The prediction and measurement of thermo¬ 
acoustic response of plate structures, 31st Structures, Structural 
Dynamics, and Material Conference, AIAA-90-0988-CP, Long Beach, 
CA, 1832-1838, Apr. 2-4,1990. 

11. Holmes, P., A nonlinear oscillator with a strange attractor, Phil. Trans., 
Roy. Soc. of London, 1979, 292A, 419-448. 

12. Dowell, E.H. and Pezeski, C., On the understanding of chaos in Duffings 
equation including a comparison with experiment, J. Appl. Mech., 1986, 
53, 5-9. 

13. Lee, J., Random vibration of thermally buckled plates: I Zero temperature 
gradient across the plate thickness, in Progress in Aeronautics and 
Astronautics, 168, Aerospace Thermal Structures and Materials for a 
New Era, Ed. E.A. Thornton, AIAA, Washington, DC, 1995. 41-67. 

14. Lee, J., Random vibration of thermally buckled plates: n Nonzero 
temperature gradient across the plate thickness, to appear in J. Vib. and 
Control, 1997. 

15. Lee, J., Large-amplitude plate vibration in an elevated thermal 
environment, Mech. Rev., 1993, 46, S242-254. 

16. Shinozuka, M. and Jan, C.-M., Digital simulation of random processes 
and its applications, J. Sound and Vib.1912, 25, 111-128. 

17. Chiu, H.M. and Hsu, C.S., A cell mapping method for nonlinear 
deterministic and stochastic systems - Part II: Examples of application, J. 
Appl. Mech., 1986, 53, 702-710. 

18. Vaicaitis, R., Nonlinear response and sonic fatigue of national aerospace 
space plane surface panels, J. Aircraft, 1994, 31, 10-18. 

19. Vaicaitis, R., Response of Composite Panels Under Severe Thermo- 
Acoustic Loads, Report TR-94-05, Aerospace Structures Information and 
Analysis Center, Wright-Patterson AFB, OH, Feb., 1994. 

20. Kasdin, N.J., Runge-Kutta algorithms for the numerical integration of 
stochastic differential equations, J. Guidance, Control, and Dynamics, 
1995,18, 114-120. 

21. Shampine, L.F. and Gordon, M.K., Computer solution of ordinary 
differential equations, 1975, Freeman, San Francisco. 

22. Lin, Y.K., Probabilistic theory of structural dynamics, Robert E. Krieger 
Publishing, 1976, Huntington, NY. 

23. Brunsden, V., Cortell, J. and Holmes, P.J., Power spectra of chaotic 
vibrations of a buckled beam, J. Sound and Vib., 1989,130, 1-25. 


917 



918 


ENHANCED CAPABELITIES OF THE NASA LANGLEY 
THERMAL ACOUSTIC FATIGUE APPARATUS 

Stephen A. Rizzi and Travis L. Turner 
Structural Acoustics Branch 
NASA Langley Research Center 
Hampton, VA 23681-0001 

ABSTRACT 

This paper presents newly enhanced acoustic capabilities of the Thermal 
Acoustic Fatigue Apparatus at the NASA Langley Research Center. The 
facility is a progressive wave tube used for sonic fatigue testing of aerospace 
structures. Acoustic measurements for each of the six facility configurations 
are shown and comparisons with projected performance are made. 

INTRODUCTION 

The design of supersonic and hypersonic vehicle stmctures presents a 
significant challenge to the airframe analyst because of the wide variety and 
severity of environmental conditions. One of the more demanding of these is 
the high intensity noise produced by the propulsion system and turbulent 
boundary layer [1]. Complicating effects include aero-thermal loads due to 
boundary layer and local shock interactions, static mechanical preloads, and 
panel flutter. Because of the difficulty in accurately predicting the dynamic 
response and fatigue of structures subject to these conditions, experimental 
testing is often the only means of design validation. One of the more common 
means of simulating the thermal-vibro-acoustic environment is through the use 
of a progressive wave tube. The progressive wave tube facility at NASA 
Langley Research Center, known as the Thermal Acoustic Fatigue Apparatus 
(TAFA), has been used in the past to support development of the thermal 
protection system for the Space Shuttle and National Aerospace Plane [2]. It 
is presently being used for sonic fatigue studies of the wing strake 
subcomponents on the High Speed Civil Transport [3]. 

The capabilities of the TAFA were previously documented by Clevenson and 
Daniels [4]. The system was driven by two Wyle WAS 3000 airstream 
modulators which provided an overall sound pressure level range of between 
125 and 165 dB and a useful frequency range of 50-200 Hz. A 360 kW quartz 
lamp bank provided radiant heat with a peak heat flux of 54 W/cm^. A 
schematic of the facility is shown in Figure 1. Representative spectra and 
coherence plots are shown in Figures 2 and 3. Since that time, the facility has 
undergone significant enhancements designed to improve its acoustic 
capabilities; the heating capabilities were not changed. The objectives of the 
enhancements were to increase the maximum overall sound pressure level 
(OASPL) to 178 dB, increase the frequency bandwidth to 500 Hz and improve 
the uniformity of the sound pressure field in the test section. This paper 


919 


documents the new capabilities of the TAFA and makes comparisons with the 
projected performance. 



Figure 1: Schematic of the old TAFA facility. 



Figure 2: Test section spectra of the Figure 3: Test section coherence of the 
old TAFA facility. old TAFA facility. 


FACILITY DESCRIPTION 

In order to meet the design objectives, extensive modifications were made to 
the sound generation system and to the wave tube itself. A theoretical increase 
of 6 dB OASPL was projected by designing the system to utilize eight WAS 
3000 air modulators compared to the two used in the previous system. A 
further increase of nearly 5 dB was expected by designing the test section to 
accommodate removable water-cooled insert channels which reduced its cross- 
sectional area from 1.9m x 0.33m to 0.66m x 0.33m. The frequency range was 
increased through the use of a longer horn design with a lower (15 Hz vs. 27 
Hz in the old facility) cut-off frequency, use of insert channels in the test 
section to shift the frequency of significant standing waves above 500 Hz, and 
design of facility sidewall stmctures with resonances above 1000 Hz. The 
uniformity of the sound pressure field in the test section was improved through 
several means. A new, smooth exponential horn was designed to avoid the 
impedance mismatches of the old design. To minimize the effect of 
uncorrelated, broadband noise (which develops as a byproduct of the sound 


920 



generation system), a unique design was adopted which allows for the use of 
either two-, four-, or eight-modulators. When testing at the lower excitation 
levels for example, a two-modulator configuration might be used to achieve a 
lower background level over that of the four- or eight-modulator 
configurations. In doing so, the dynamic range is extended. Lastly, a catenoidal 
design for the termination section was used to smoothly expand from the test 
section. 

Schematics of the facility in the three full test section configurations are shown 
in Figures 4-6. In the two-modulator configuration, the 2 x 4 transition cart 
acts to block all but two of the eight modulators. The facility is converted 
from the two- to four-modulator configuration by the removal of the 2 x 4 
transition cart and connection of two additional modulators. In doing so, the 
modulator transition cart slides forward and thereby maintains the continuous 
exponential expansion of the duct. In the four-modulator configuration, the 4 
X 8 transition cart acts to block the two upper and two lower modulators. 
Removal of this component and connection of the four additional modulators 
converts the facility to the eight-modulator configuration. Again, the 
continuous exponential expansion is maintained as the modulator transition 
cart slides forward. 



Figure 4: Two-modulator full test section configuration. 


921 














Figure 5: Four-modulator full test section configuration. 



Figure 6: Eight-modulator full test section configuration. 


Schematics of the three reduced test section configurations are shown in 
Figures 7-9. In these configurations, the horn cart is discarded and the horn 
transition cart mates directly to the test section. Water-cooled inserts are used 
in the test section to reduce its cross-sectional area. Upper and lower inserts in 
the termination section are used to smoothly transition the duct area to the full 
dimension at the exit. Conversion from the two- to the four-modular 
configuration and from the four- to the eight-modulator configuration is again 
accomplished through removal of the 2 x 4 and 4x8 transition carts, 
respectively. 


922 













HORN TTUNSTTION SECTION 



Figure 7: Two-modulator reduced test section configuration. 



Figure 8: Four-modulator reduced test section configuration. 

TEST PROCEDURE 


Measurements were taken for several conditions in each of the six facility 
configurations. Each modulator was supplied with air at a pressure of 207 kPa 
(mass flow rate of approximately 8.4 kg/s) and was electrically driven with the 
same broadband (40-500 Hz) signal. Acoustic pressures were measured at 
several locations along the length of the progressive wave tube using B&K 
model 4136 microphones and Kulite model MIC-190-HT pressure transducers, 
see Table 1. The positive x-direction is defined in the two-modulator full 
configuration (from the modulator exit) along the direction of the duct. The 
positive y-direction is taken vertically from the horizontal centerline of the 


923 
















HORN TRIWSmOH SECTION 


MODULATOR TRANSITION 
FLEMSie HOSE 


adapter puts assembly 

TEST SECTION 

ADAPTER plate assembly 
TERM1ASAT10N SECTION 



horn TRANSITION CART 


Figure 9: Eight-modulator reduced test section configuration. 

duct and the positive z-direction is defined from the left sidewall of the duct as 
one looks downstream. 


Table 1: Kulite (K) and microphone (M) locations of acoustic measurements. 


Loc. 

Description 

Type 

Coordinate (m) 

1 

Test Sect. Horizontal Centerline Upstream 

K 

7.75, 0, 0 

2 

Test Sect. Horizontal Centerline Downstream 

K 

8.71, 0, 0 

5 

Test Sect. Vertical Centerline Top 

M 

8,23, 0.3, 0 

15 

Test Sect. HorizontaWertical Centerline 

K 

8,23,0, 0 

25 

Test Sect. Vertical Centerline Bottom 

M 

8.23, -0.3, 0 

28 

2x4 HorizontaWertical Centerline 

M 

2.19, 0,0 

29 

4x8 Horizontal Centerline, % Downstream 

M 

3.66,0, 0 

30 

Horn Tran, Hor. Centerline, % Downstream 

M 

4.75,0, 0 

35 

Termination HorizontaWertical Centerline 

M 

12.46,0, 0.17 


The acoustic pressure at location 1 was used as a reference measurement for 
shaping the input spectrum and for establishing the nominal overall sound 
pressure level for each test condition. For each configuration, the input 
spectrum to the air modulators was manually shaped through frequency 
equalization to produce a nearly flat spectrum at the reference pressure 
transducer. Data was acquired at the noise floor level (flow noise only) and at 
overall levels above the noise floor in 6 dB increments (as measured at the 
reference location) up to the maximum achievable. Thirty-two seconds of 
time data were collected at a sampling rate of 4096 samples/s for each 
transducer in each test condition. Post-processing of the time data was 
performed to generate averaged spectra and coherence functions with a 1-Hz 
frequency resolution. 


924 







RESULTS 

For each facility configuration, plots of the following quantities are presented: 
normalized input spectrum to the air modulators, minimum to maximum 
sound pressure levels at the reference location, maximum sound pressure 
levels in the test section, maximum sound pressure levels upstream and 
downstream of the test section, and vertical and horizontal coherence in the 
test section. The minimum levels in each case correspond to the background 
noise produced by the airflow through the modulators. 

Normalized input voltage spectra to each modulator for each configuration are 
shown in Figures 10, 15, 20, 25 and 30. These spectra were generated to 
achieve as flat an output spectrum as possible at the reference location for the 
frequency range of interest (40-200 Hz for the full section, 40-500 Hz for the 
reduced section). As expected, the significant difference between the full and 
reduced configurations is seen in the high (>200 Hz) frequency content. 

Figure 11 shows a background noise level of 126 dB (the lowest of all 
configurations) for the two-modulator full test section configuration. Nearly 
flat spectra are observed below 210 Hz for levels above 130 dB, giving a 
dynamic range of about 32 dB. The flat spectrum shape is a significant 
improvement over the performance of the old configuration as shown in Figure 
2. Standing waves are evident at frequencies of 210, 340 and 480 Hz. For this 
reason, the full section operation is limited to less than 210 Hz or to the 220- 
330 and 370-480 Hz frequency bands. The effect of standing waves are 
explored in further depth in the next section. The spectra in Figure 12 indicate 
a nearly uniform distribution in the x-direction throughout the test section. It is 
interesting to note that Figure 13 shows no sign of standing waves upstream of 
the test section, confirming that the cause is associated with the test section. 
Lastly, a near perfect coherence between upstream and downstream, and upper 
and lower test section locations is shown in Figure 14 for frequencies between 
40 and 210 Hz. Again, this is a significant improvement over the performance 
of the old configuration (Figure 3). 



Figure 10: Normalized input spectrum Figure 11: Min to max SPL at location 
(2-modulator full). 1 (2-modulator full). 


925 



Figure 12: SPL in test section at max Figure 15: Normalized input spectrum 
level (2-modulator full). (4-modulator full). 



Figure 13: SPL along length of TAFA Figure 16: Min to max SPL at location 
(2-modulator full). 1 (4-modulator full). 



Figure 14: Test section coherence (2- Figure 17: SPL in test section at max 
modulator full). level (4-modulator full). 

The four-modulator full configuration exhibits similar behavior as the two- 
modulator full configuration as seen in Figures 16-19. The lowest level at 
which a uniform spectrum is achieved is 137 dB, giving a dynamic range of 
roughly 30 dB in this configuration. Lastly, the eight-modulator full 


926 











configuration results, shown in Figures 21-24, indicate a noise floor of about 
142 dB and dynamic range of 22 dB. 



Frequency, Hz Frequency, Hz 


Figure 18: SPL along length of TAFA Figure 21: Min to max SPL at location 
(4-modulator full). 1 (8-modulator full). 



Frequency, Hz Frequency, Hz 

Figure 19: Test section coherence (4- Figure 22: SPL in test section at max 
modulator full). level (8-moduiator full). 



Frequency, Hz Frequency, Hz 


Figure 20: Normalized input spectrum Figure 23: SPL along length of TAFA 
(8-modulator full). (8-modulator full). 


927 













Frequency, Hz Frequency, Hz 


Figure 24: Test section coherence (8- Figure 27: SPL in test section at max 
modulator full). level (2-modulator reduced). 



Figure 25: Normalized input spectrum Figure 28: SPL along length of TAFA 
(2-modulator reduced). (2-modulator reduced). 



Figure 26: Min to max SPL at location Figure 29: Test section coherence (2- 
1 (2-modulator reduced). modulator reduced). 

The reduced test section configurations are used to increase the frequency 
range and maximum sound pressure level in the test section. Results for the 
two-modulator reduced configuration, shown in Figures 26-29, indicate a 
nearly flat spectrum between 40 and 480 Hz, a noise floor of 129 dB and a 
dynamic range of about 28 dB. Coherence in the test section is nearly unity 


928 











over this frequency range. This represents a significant improvement over the 
old facility configuration. Results of similar quality indicate a d 5 mamic range 
of roughly 26 and 29 dB for the four- (Figures 31-34) and eight-modulator 
(Figures 36-39) configurations, respectively. Note that the coherence for these 
configurations is slightly reduced at the high frequencies, but is still very good 
out to 480 Hz. 



Figure 30: Normalized input spectrum Figure 33: SPL along length of TAFA 
(4-modulator reduced). (4-modulator reduced). 


160 r 


80 


OASPLs: 134.1, 135.5, 142.1, 147.6, 
153.9, 160.1, 165.9,167.9 


1.0 



! 1 



1 1 


Loc 1, Loc 2 
Loc 5, Loc 25 


400 


100 


200 300 

Frequency, Hz 


400 


500 


Figure 31: Min to max SPL at location Figure 34: Test section coherence (4- 
1 (4-modulator reduced). modulator reduced). 



Figure 32: SPL in test section at max Figure 35: Normalized input spectrum 
level (4-modulator reduced). (8-modulator reduced). 


929 








80 


OASPLs; 134.1,135.7, 141.9,148.3, 
154.1. 160.0, 165.9.170.5 


80 


Loc30 

Loci 


100 200 300 400 500 100 200 300 400 500 

Frequency. Hz Frequency, Hz 

Figure 36: Min to max SPL at location Figure 38: SPL along length of TAFA 
i (8-modulator reduced). (8-modulator reduced). 



Figure 37: SPL in test section at max Figure 39: Test section coherence (8- 
level (8-modulator reduced). modulator reduced). 


Table 2 presents a summary of the maximum average OASPL for each facility 
configuration. In each case, the number of active modulators were run at 
maximum power as an independent group (independently for the single 
modulator case) and the results averaged. For example, results for one active 
modulator were obtained by running each modulator individually and 
averaging the resulting pressures. 


Table 2: Summary of maximum average overall sound pressure levels (dB). 



Number of Active Modulators 


1 

2 

4 

8 

2-Modulator Red. 



1 

1 j 

2-Modulator Full 



i 


4-Modulator Red. 

159.1 

— 

MSM 


4-Modulator Full 

155.6 

161.2 



8-Modulator Red. 

158.4 

— 

mmm 

171.7* 

8-Modulator Full 

153.0 

158.4 

164.5 

170.0 

^Pressure scaled by ^7? from 7-modu 

ator run 

















DISCUSSION 


In this section, limiting behaviors of the full and reduced test section 
configurations are explored and the effect of test section inserts, modulator 
coupling and wave tube performance are discussed. 

Limiting Behaviors 

The auto-spectra from the full test section configurations exhibit sharp 
reductions in level at approximately 210, 340, and 480 Hz. This behavior 
corresponds to measurements near nodes of vertical (height) standing waves in 
the test section portion of the wave tube. Table 3 summarizes theoretical, 
resonant frequencies and corresponding modal indices of the test section duct 
resonances within the excitation bandwidth. The modal indices m and n 
correspond to half wavelengths in the vertical and transverse (width) directions 
of the cross section, respectively. There are several resonances that may be 
excited below 500 Hz, but only three of these appear to be significant at the 
test section transducer locations (about the horizontal centerline). Because of 
the presence of air flow in the facility and lack of measurements in the cross 
section, it is difficult to correlate the experimental and theoretical modes. 
Measurements of the acoustic pressure at several locations in a cross-section of 
the duct will be necessary to fully characterize the resonant behavior. It is 
sufficient to say that the usable frequency range in the full test section 
configurations is approximately 40-210 Hz near the horizontal centerline. 
Acoustic pressure auto-spectra from the reduced test section configurations are 
essentially flat to almost 500 Hz. This is due to the fact that only two 
resonances are within the excitation bandwidth for this configuration, see 
Table 3. A sharp reduction is noted in the vicinity of 480 Hz. Although the 
(m=l, n=0) resonance does not appear to be significant, close inspection of the 
data (not shown) indicates its presence. Therefore, the usable frequency range 
for the reduced test section configurations is approximately 40-500 Hz. 

Table 3: Theoretical resonant frequencies of test section duct modes in Hz. 











Performance of Test Section Configurations 

For constant input acoustic power, the change from full to reduced test section 
configurations should theoretically result in a 4,7 dB increase in OASPL. 
However, Table 2 shows that increases of only 2.1 (e.g. 164.3-162.2), 0.9, and 
1.7 dB were realized for the two-, four- and eight-modulator configurations. 
The system efficiency (actual/expected mean-square pressure) of the two-, 
four- and eight-modulator reduced configurations is 38, 40 and 44 percent, 
respectively, compared with 51, 63 and 62 percent for the two-, four- and 
eight-modulator full configurations. The expected pressure is calculated based 
upon a input-scaled value of the rated acoustic power of the WAS 3000 
modulator assuming incoherent sources (3 dB per doubling). In general, the 
full section efficiency is greater than the corresponding reduced section 
efficiency. While the reason for this phenomena is not known, it is 
conjectured that the lack of expansion in the reduced configurations limits the 
development of plane waves. Therefore, phase and amplitude mismatches 
between acoustic sources may be accentuated. 

Modulator Coupling Performance 

A simplified waveguide analysis for coherent, phase-matched sources predicts 
increases in OASPL as shown in Table 4. Measured performance gains were 
less than predicted because of the assumptions of the waveguide analysis 
(inactive source area treated as hard wall), and possible reductions due to 
phase differences between modulators and non-parallel wave fironts at the exit 
of the modulator cart, see Figures 4-9. The latter effect is due to different 
angles of inclination of the sources relative to the axis of the wave tube. The 
greater gains achieved in the full test section configurations support the above 
contention that they are more efficient than the reduced configurations in 
combining the acoustic sources. 


Table 4: Change in SPL (dB) from 1 to max. number of active modulators. 


Configuration 

A SPL fi:om 1 Active Mod. (Meas/Pred) 

2-Modulator Red. (2 active mods.) 

3.9/6.53 

2-Modulator Full (2 active mods.) 

5.5/6.53 

4-Modulator Red. (4 active mods.) 

8.8 /13.98 

4-Modulator Full (4 active mods.) 

11.4/13.98 

8-Moduiator Red. (8 active mods.) 

13.3/22.10 

8-Moduiator Full (8 active mods.) 

17.0/22.10 


Wave Tube Performance 


A change in configuration from the two- to the four-modulator configurations, 
and from the four- to the eight-modulator configurations, will result in an 
incremental increase of 3 dB in OASPL if the individual sources are phase- 
matched. This is due to a pure doubling of the power without any change in 
the radiation impedance of the individual sources. For the reduced 
configurations, a 3.6 and 3.8 dB increase are observed, respectively. A 4.8 and 


932 








3.0 dB increase are observed for the full configurations, respectively. Note 
that a greater than 3 dB increase is possible when the higher modulator 
configuration (for example, the four-modulator reduced configuration) is less 
susceptible than the lower modulator configuration (the two-modulator 
reduced configuration) to phase mismatches between modulators. This seems 
plausible because any such mismatches are averaged over a larger number of 
sources. 

SUMMARY 

Modifications to the NASA Langley TAFA facility resulted in significant 
improvements in the quality and magnitude of the acoustic excitation over the 
previous facility. The maximum OASPL was increased by over 6 dB (vs the 
previous 165 dB) with a nearly flat spectrum between 40-210 and 40-480 Hz 
for the full and reduced test section configurations, respectively. In addition, 
the coherence over the test section was excellent. These improvements, 
however, did not meet the objective for a maximum OASPL of 178 dB. 

There are several reasons why the maximum OASPL did not meet the 
objectives, including a lack of expansion in the reduced configurations and 
phase differences between modulators. A detailed computational analysis 
would be desirable to indicate the source of the inefficiencies and to help 
identify possible means of increasing the overall system performance. 
ACKNOWLEDGEMENTS 

The authors wish to thank Mr. H. Stanley Hogge and Mr. George A. Parker for 
their support in configuring and running the facility. We wish to also thank 
Mr. James D. Johnston, Jr. of NASA Johnson Space Center for loan of four 
Wyle air modulators. 

REFERENCES 

1. Maestrello, L., Radiation from a Panel Response to a Supersonic 
Turbulent Boundary Layer, Journal of Sound and Vibration, 1969, 
10(2), pp. 261-295. 

2. Pozefsky, P., Blevins, R.D., and Langanelli, A.L., Thermal-Vibro- 
Acoustic Loads and Fatigue of Hypersonic Flight Vehicle Structure, 
AFWAL-TR-89-3014, 

3. Williams, L.J., HSCT Research Gathers Speed, Aerospace America, 
April 1995, pp. 32-37. 

4. Clevenson, S.A. and Daniels, E.F., Capabilities of the Thermal 
Acoustic Fatigue Apparatus, NASA TM 104106, February 1992. 


933 




SONIC FATIGUE CHARACTERISTICS OF HIGH TEMPERATURE MATERIALS AND 
STRUaURES FOR HYPERSONIC FLIGHT VEHICLE APPLICATIONS- 


Dr. I. Holehouse, Staff Specialist, 
Rohr Inc., Chula Vista, California 


1. INTRODUCTION SUWiARY 

A combined analytical and experimental program was conducted to investigate 
thermal-acoustic loads, structural response, and fatigue characteristics of 
skin panels for a generic hypersonic flight vehicle. Aerothermal and 
aeroacoustic loads were analytically quantified by extrapolating existing 
data to high Mach number vehicle ascent trajectories. Finite-element 
thermal and sonic fatigue analyses were performed on critically affected 
skin panels. High temperature random fatigue shaker tests were performed 
on candidate material coupons and skin-stiffener joint subelements to 
determine their random-fatigue strength at high temperatures. These were 
followed by high temperature sonic fatigue tests of stiffened-skin panels 
in a progressive wave tube. The primary materials investigated were 
carbon-carbon and silicon-carbide refractory composites, titanium metal 
matrix composites and advanced titanium alloys. This paper reports on the 
experimental work and compares measured frequencies and acoustically 
induced response levels with analytically predicted values. 

The coupon shaker test data were used to generate material random fatigue 
"S-N" curves at temperatures up to 980°C. The joint subelements provided 
data to determine the effects on fatigue life of skin-stiffener joining 
methods. The PWT sonic fatigue panel tests generated response and fatigue 
life data on representative built-up skin panel design configurations at 
temperatures up to 925“C and sound pressure levels up to 165 dB. These^ 
data are used in determining the response strains and frequencies of skin 
panel designs when subjected to combined thermal-acoustic loading and to 
identify modes of failure and weaknesses in design details that affect_ 
sonic fatigue life. Sonic fatigue analyses of selected test panel design 
configurations using finite-element techniques were also performed and 
related to the experimental results. Acoustically induced random stresses 
were analytically determined on a mode-by-mode basis using finite element 
generated mode shapes and an analytical procedure that extends Miles 
approach to include multi-modal effects and the spatial characteristics of 
both the structural modes and the impinging sound field. 

The paper also describes the instrumentation development work performed in 
order to obtain reliable strain measurements at temperatures in excess of 
conventional strain gauge capabilities. This work focused primarily on the 
use of recently developed high temperature (350"C to 1000“C) strain gauges, 
laser Doppler vibrometers, high temperature capacitance displacement 
probes, and the determination of strain-displacement relationships to 
facilitate the use of double integrated accelerometer data to derive strain 
levels. 


935 


This work was funded by the USAF Flight Dynamics Laboratory (Kenneth R. 
Wentz, Project Engineer). The complete program report is contained in 
References 1 and 2. 


2. HIGH TEMPERATURE STRAIN MEASUREMENTS 

Conventional adhesively bonded strain gauge installations are temperature 
limited to approximately 350°C. In order to achieve strain measurements at 
higher temperatures, up to QSO^C, ceramic layers and coatings were used to 
both attach strain gauges and to thermally protect them. However, such 
strain gauge installations are very sensitive to process parameters which 
often need varying depending upon the test specimen material. Coated 
carbon-carbon is a particularly difficult material to adhere to due to its 
material characteristics and relatively rough surface texture. Carbon- 
carbon also has a near zero coefficient of thermal expansion which presents 
attachment and fixturing problems in a high temperature environment. 

When high test temperatures either preclude or make problematic the use of 
strain gauges, an alternative technique for obtaining strain levels is to 
measure displacements and then determine strain levels using strain- 
displacement ratios. Strain is directly proportional to displacement for a 
given deflected shape, or mode shape, regardless of changes in the elastic 
modulus of the specimen material as it is heated. Consequently, if the 
deflected shape does not change significantly with temperature, high 
temperature test strain levels can be determined from room temperature 
strain and displacement measurements in combination with displacement 
measurements made at the test temperature. 

This measurement technique facilitates the use of non-contacting 
transducers which can be located away from the heated area, such as 
capacitance displacement probes or Laser Doppler Vibrometers (LDV). LDVs 
actually measure surface velocity but their signal outputs can be readily 
integrated and displayed as displacement. Accelerometers can also be used 
to measure displacement by double integrating their signal output. 

However, since accelerometers require surface contact they have to either 
withstand, or be protected from, the thermal environment. When this is not 
readily achievable, it is sometimes possible to install an accelerometer at 
a location on the test specimen or fixturing where the temperature is 
within its operating range, providing the displacement response at the 
point of measurement is fully coherent with the strain response at the 
required location. 

The displacement range limitations of the LDV and capacitance probes 
available to the program resulted in having to use double-integrated 
accelerometer outputs to measure displacements at room temperature and at 
the test temperature. Conventional strain gauges were used to measure 
strains at room temperature. In order to confirm that the strain- 
displacement ratios were unaffected by temperature, limited high 
temperature strains were measured at temperatures up to 980*C. Once the 
strain-displacement ratio for a given specimen type was determined, air¬ 
cooled accelerometers were used to determine high temperature test strain 
levels. The level of measurement accuracy of this technique was estimated 
to be within 10 percent. 


936 



The most successful strain measurements made at 980“C utilized a ceramic 
flame spray installation of an HFN type free filament gauge. This gauge 
installation included the use of silicon-carbide (SiC) cement as a base 
coat for the gage, applied over a 1-inch square area of a lightly sanded 
carbon-carbon surface substrate. Lead wire attachments to the gauge were 
made with standard Nichrome ribbon wire anchored to the specimen with SiC 
cement. With this gauge installation, it was possible to make dynamic 
strain measurements for short periods of time at 980“C. 


3. RANDOM FATIGUE SHAKER TESTS 

The instrumented test specimens were mounted in a duckbill fixture and the 
specimen/fixture assembly then enclosed in a furnace. An opening in the 
furnace allows the specimen tip to protrude out in order to accorrmodate the 
air-cooled tip accelerometer. Figure 1 shows strain gauge locations and 
fixturing for material coupon and joint subelement specimens. 

The test procedure comprised a room temperature sine-sweep in order to 
identify the fundamental mode and its natural frequency, one-third octave 
random loading at room temperature centered around the fundamental natural 
frequency and one-third octave random endurance testing at the required 
test temperature and load level. 

Twelve inhibited carbon-carbon material coupons generated usable S-N data, 
eleven at 980“C and one at 650“C. S-N data points were also generated at 
980°C for two integral joint and two mechanically fastened joint 
subelements. Fixturing problems and specimen availability limited the 
number of S-N data points generated. Figure 2 shows the random fatigue S-N 
data points with joint subelement data points superimposed. The random 
fatigue endurance level for the material coupons, extrapolated from 10 to 
10® cycles, is approximately 320 microstrain rms. The integral joint 
subelements did not fail at the strain gauge locations; consequently, the 
actual maximum strain levels were higher than those shown on Figure 2. 
Taking this into account, it appears that the integral joints have a 
fatigue endurance level of greater than one-half of that for the material 
coupons. The mechanically fastened joint subelements exhibited fatigue 
strength comparable to that of the material coupons. These results 
indicate that carbon-carbon joints and attachments methods are not 
critically limiting factors in the structural applications of inhibited 
carbon-carbon. Figure 3 shows a representative example of the strain 
amplitude and peak strain amplitude probability density functions at room 
temperature for a material coupon specimen. The "peak" function can be 
seen to approximate a Rayleigh distribution, as it should for a Gaussian 
random process. 

Random fatigue S-N data were also generated for enhanced silicon-carbide 
composites (SiC/SiC) including thermally exposed specimens (160 hours at 
980“C), titanium metal matrix composites (TMC) utilizing Ti 15-3 and Beta 
21S titanium matrix materials, titanium aluminide (super alpha two), 
titanium 6-2-4-2, titanium 6-2-4-2-$i (including thermally exposed 
specimens) and Ti-1100. The fatigue endurance levels are shown in Table 1. 
Also shown in Table 1 are S-N data points for uninhibited carbon-carbon 
generated on a previous program (Reference 3). 


937 


TABLE 1. SUMMARY OF RANDOM FATIGUE ENDURANCE LEVELS. 


material 

TEMPERATURE 

ENDURANCE LEVEL CORRESPONDING TO 10® 
CYCLES: OVERALL RMS STRAIN (MICROSTRAIN) 


material COUPONS 

SUBELEMENTS 

INHIBITED CARBON-CARBON 

1800“F (980"C) 

320 

integral JOINTS > 160 

BOLTED JOINTS 320 


5 PLY 

1800T (980"C) 

100 

- 

UNINHlbl1tU 
CARBON-CARBON 

1ft PI Y 

ISOO'-F igSOT) 

150 

_ 


*5 PLY 

1000‘'F I540"C) 

100 

- 

18 PLY 

lOOO'F (540-C) 

450 

- 

5 PLY 

ROOM TEMPERATURE 

550 

- 

18 PLY 

ROOM TEMPERATURE 

450 

- 

ENHANCED 

SiC/SiC 

NON-EXPOSED 

1800"F r980“C) 

450 

- 

THERMALLY EXPOSED 

iaOO“F f980‘'C) 

300 

- 



1000“F (540*0 

520 

DIFFUSION-BONDED HAT- 
STIFFENED = 520 

n IML 

ROOM TEMPERATURE 

2250 

- 

beta 21S TMC 


200 



510 

- 

TITANIUM ALUMINIDE 

/ciinCD ftl DUIA TUn^ 

ROOM TEMPERATURE 

410 


{ rvui 1 iri • * 

Ti-6242-Si 



735 


Hilllll 


LIO BONDED HONEYCOMB 
BEAM = 388 


■■ 


WELDED JOINT = 400 

T-; nnn 




- 

Ti 6-2-4-2 

ROOM TEMPERATURE 

675 

- 


866MISC/039-T1.IH 

12-02-96 


938 










Figure 4 shows random fatigue S-N curves for the materials tested 
superimposed on one graph for comparison purposes. The Ti 15-3 TMC data 
are not shown since this was a concept demonstrator material utilizing a 
Ti 15-3 matrix material for producibility reasons. Ti 15-3 does not have 
the temperature capability for hypersonic vehicle applications. Titanium 
aluminide data are not shown due to its brittle material characteristics 
making it unsuitable for sonic fatigue design critical structures. Ti-1100 
S-N data were very similar to the non-exposed Ti 6-2-4-2-Si and are not 
shown. Ti 6-2-4-2 coupons were only tested at room temperature before 
beingVeplaced by Ti 6-2-4-2-Si, which has higher structural temperature 
capabilities. 

The fatigue curves in Figure 4 show inhibited carbon-carbon to have higher 
fatigue strength at 980°C than does its uninhibited counterpart. Inhibited 
carbon-carbon also has greater resistance to oxidation at high 
temperatures. 

Although unexposed enhanced SiC/SiC had greater random fatigue strength at 
980“C than did inhibited carbon-carbon, the two materials exhibited similar 
strength at temperature after allowing for thermal exposure. However, 
SiC/SiC has a maximum temperature capability of 1100 to 1200°C compared to 
1700 to 1900“C for carbon-carbon. 

The Beta 21S TMC material demonstrated resonable fatigue strength at 815°C 
and the Ti 6-2-4-2-Si specimens exhibited high fatigue strength at 620°C to 
650“C. 


4. SONIC FATIGUE PANEL TESTS 

These tests were performed in Rohr's high temperature progressive-wave tube 
(PWT) test facility. The facility is capable of generating overal1^sound 
pressure levels of 165 to 168 dB at temperatures up to 925“C to 980“C, 
depending upon the test panel configuration and material. 

Three rib-stiffened carbon-carbon panels and a monolithic hat-stiffened 
Beta 21S TMC panel were subjected to sonic fatigue testing. Response 
strains were measured on the four panels over a range of incrementally 
increasing sound pressure levels (140 to 165 dB) at room temperature. One 
carbon-carbon panel was subjected to sonic fatigue testing at room 
temperature and the other two tested at 925“C. The TMC panel was endurance 
tested at 815“C. Figures 5 and 6 show a carbon-carbon panel and its 
fixturing installed in the PWT. The panels were attached to the fixture 
via flexures in order to allow for differences in the thermal expansion of 
the panel and fixture materials. Structural details of the panels and 
instrumentation locations are given in References 1 and 2. 

The three carbon-carbon panel configurations encompassed two skin 
thicknesses and two stiffener spacings as follows: 

Panel 1: 3 skin bays, 6 in. by 20 in. by 0.11 in. thick 

Panel 2: 2 skin bays, 9 in. by 20 in. by 0.11 in. thick 

Panel 3: 3 skin bays, 6 in. by 20 in. by 0.17 in. thick 


939 



Table 2 summarizes the measured room temperature frequencies and strain 
response levels: 


TABLE 2. ROOM TEMPERATURE RESPONSE OF TEST PANELS- 


TEST 

PANEL 

FREQUENCY OF 
IN-PHASE 
MODE (Hz) 

OASPL 

(dB) 

OVERALL RMS STRAINS 
rMICROSTRAIN) 

EDGE OF 
SKIN BAY 

CENTER OF 
SKIN BAY 

CARBON-CARBON NO. 1 

267 

165 

305 

149 


155 & 171 

145 

126 

39 




191 

59 



165 

558* 

173* 

CARBON-CARBON NO. 3 

423 

165 

69 

127 

BETA 21$ TMC 

241 

165 

HIGHEST STRAIN = 287 




AT PANEL CENTER ON 




STIFFENER CAP 


* EXTRAPOLATED ON THE BASIS OF TUE STRAIN RESPONSE WITH SPL FOR 
PANELS 1 AND 3. 


Panel 1 was subjected to 165 dB at room temperature for 10 hours at which 
point cracks developed at the ends of the stiffeners. The frequency 
dropped slightly during the ten hour test resulting in the number of cycles 
to failure being approximately 9 million. 

Panel 2 was endurance tested at 925“C at 150, 155 and 160 dB for 3-1/2 
hours at each level, followed by one hour at 165 dB. At this point, cracks 
were observed at the ends of the stiffeners, similar to the cracks in 
Panel 1. 

Panel 3 was endurance tested at 925“C and 165 dB for 10 hours without any 
damage to the panel. 

The TMC panel was endurance tested at 815°C and 165 dB for 3-1/2 hours at 
which time cracks were observed in two stiffener caps at the panel center. 

The high test temperatures for Panels 2 and 3 and the TMC panel precluded 
attaching an accelerometer directly to the panel surface, even with air 
cooling. This prevented the direct measurement of panel displacements at 
925°C. In order to attempt to estimate the high temperature endurance test 
strain levels, a temperature survey was performed on the panel fixturing 
with Panel 3 installed in order to determine an acceptable location for an 
accelerometer. An accelerometer at the selected fixture location tracked 
linearly with the highest reading strain gauges during a room temperature 
response survey. The coherence between the fixture accelerometer and the 
panel strain gauges was 0.9 in the frequency range of panel response. 


940 



Having established a coherent strain displacement relationship at room 
temperature, the temperature was increased progressively with increasing 
acoustic loading, generating accelerometer and microphone data at 480 C and 
140 dB, 650°C and 155 dB, 860°C and 155 dB, and 925“C at 165 dB. It was 
clear from the data at the higher temperatures and load levels that the 
full spectrum overall rms displacement levels obtained by double integrat¬ 
ing the accelerometer output signals could not be used to determine high 
temperature strain levels due to high amplitude, low frequency displace¬ 
ments (displacement being inversely proportional to frequency squared for a 
given "g" level) that were well below the panel response frequency range 
and therefore would not be proportional to panel strain levels. It is 
important to remember here that since the accelerometer is mounted on the 
panel fixture, it is measuring fixture response, some of which is not 
related to panel response. 


After reviewing the various frequency spectra, it was decided to re-analyze 
the data to generate overall rms levels over selected frequency bandwidths 
that would encompass a high percentage of the full-spectrum overall rms 
strains and eliminate the low frequency displacements. If a consistent 
strain-displacement relationship could be established at room temperature 
within a frequency bandwidth such that the strains could be related to the 
full-spectrum overall rms strains, and if the same bandwidth could be used to 
generate displacements at temperatures that were sufficiently consistent to 
relate to strain response, then it would be possible to at least make a 
reasonable estimate of the test temperature strain level. It was determined 
that band-passed response data in the 300 to 600 Hz frequency range gave 
consistent strain-displacement ratios at room temperature. Double-integrated 
band-passed accelerometer outputs (displacements) were consistent with 
increasing sound pressure levels at incrementally increasing test temperatures 
up to the 925°C/165 dB endurance test conditions. Table 3 summarizes the high 
temperature test panel results. 


TABLE 3. HIGH TEMPERATURE TEST PANEL RESULTS. 


TEST PANEL 

TEST 

TEMPERATURE 

OVERALL 
SOUND 
PRESSURE 
LEVEL (dB) 

HIGHEST 
ESTIMATED 
OVERALL RMS 
STRAIN 

(MICROSTRAIN) 

EXPOSURE TIME, 
ESTIMATED FATIGUE 

CYCLES AND COMMENTS 

(°F) 

rc) 

BETA 21S TMC 
PANEL 

1500 

815 

165 

NOT ESTIMATED 

3 1/2 HRS, 3x10“ CYCLES, 
STIFFENERS CRACKED AT 
MID-SPAN 

CARBON-CARBON 
PANEL NO. 2 



150 

155 

TTTORSTTTMo^TYOlsr 

NO FAILURE 

155 

219 ^ 

3 1/2 HRS, 2.3x10° CYCLtS, 
NO FAILURE 

160 

316 

3 1/2 HRS. 2.3x10“ CYCLES, 
NO FAILURE 

165 

453 

1 HR, 6.4xl0‘> cycles! 

CRACKS AT STIFFENER ENDS 



925 

165 

103 

10 HRS, 1.7x10' CYCLES, 

NO FAILURE 


941 















It should be noted that carbon-carbon panels 1 and 2 exhibited cracks at 
the stiffener ends, whereas the maximum measured strains were at the edges 
of skin bays. Consequently, the actual strain levels at the crack 
locations were either higher than the measured levels or there were 
significant stress concentrations at the stiffener terminations. 


5. COMPARISON OF ANALYTICAL AND TEST RESULTS FOR CARBON-CARBON PANELS 

MSC NASTRAN was used to perform finite element analyses on the three 
carbon-carbon panels that were subjected to the sonic fatigue testing 
described in Section 4. The oxidation resistant coating was modeled as a 
non-structural mass, which is compatible with the panel test results.^ 
Natural frequencies, mode shapes and acoustically induced random strain 
levels were analytically determined for room-temperature conditions and 
compared to the room-temperature panel test results. 

Acoustically induced random stresses were analytically determined on a 
mode-by-mode basis using the finite element generated mode shapes and a 
Rohr computer code based on an analytical procedure presented in 
Reference 4. This procedure extends Miles* approach (Reference 5) to 
include multi-modal effects and the spatial characteristics of both the 
structural modes and the impinging sound field. 

Table 4 shows the calculated and measured frequencies, overall rms strain 
levels and the strain spectrum levels for the in-phase stiffener bending 
mode for the carbon-carbon panels at room temperature. 


TABLE 4. CALCULATED AND MEASURED RESPONSE FREQUENCIES AND STRAIN 
LEVELS FOR CARBON-CARBON PANELS AT ROOM TEMPERATURE. 




STRAIN LEVELS AT EDGE OF SKIN BAY 

NATUF 
FREQUE 
OF IN-F 
MODE 
(Hzl 

\fKL 

:ncy 

>HASE 

OVERALL RMS STRAIN 
(MICROSTRAIN) 

STRAIN 

SPECTRUM LEVEL 
IN-PHASE MODE 
(MICROSTRAIN/Hz) 

FE ANALYSIS 

MEASURED 

FE ANALYSIS 

MEASURED 

FE ANALYSIS 

MEASURED 

PANEL 1 
a65 dBl 

305 

267 

510 

305 

84 

60 

PANEL 2 
(145 dBl 

190 

155 & 
171 

133 

126 

40 

41 

PANEL 3 
(165 dB) 

460 

423 

77 

69 

16 

16 


942 
















The above results show good agreement between the finite element generated 
values and those measured. The level of agreement is particularly good for 
the strain spectrum levels, which are typically more difficult to 
accurately predict. Figure 7 shows the finite-element frequency solution 
for Panel 3. The in-phase mode shape can be seen to have an overall modal 
characteristic due to the relatively low bending stiffness of the 
stiffeners for the skin thickness used. Figure 8 shows the measured and 
finite-element generated strain frequency spectra for Panel 3. 

Details of the finite-element analyses and models are contained in 
References 1 and 2. 


6. CONCLUSIONS AND RECOMMENDATIONS 

1. The high temperature testing techniques and strain measuring 
procedures successfully generated usable random fatigue S-N 
curves and panel response data. The use of strain-displacement 
ratios were shown to be an effective alternative to high 
temperature strain gauge measurements. 

2. In general, the materials and structural concepts tested 
demonstrated their suitability for hypersonic flight vehicle skin 
panel applications. The major exception was Titanium-Aluminide 
Super Alpha Two which was determined to be too brittle. 

3. Inhibited carbon-carbon exhibited significantly higher random 
fatigue strength at 980°C than did the uninhibited carbon-carbon 
— two to three times the random fatigue endurance strain level. 

4. Thermally exposed enhanced SiC/SiC had comparable fatigue 
strength to that of inhibited carbon-carbon at 980°C. 

5. The TMC specimens usefully demonstrated the fatigue strength of 
the TMC concept and the need to develop the concept to 
incorporate higher temperature capability titanium matrix 
materials. 

6. Titanium 6-2-4-2-Si exhibited high fatigue strength in the 590°C 
to 650“C temperature range and also demonstrated the need for TMC 
materials to utilize higher temperature matrix materials in order 
to be cost effective against the newer titanium alloys. 

7. The level of agreement between the finite element analysis 
results for the carbon-carbon panels and the progressive-wave 
tube test data demonstrated the effectiveness of the analytical 
procedure used. The analysis of structures utilizing materials 
such as carbon-carbon clearly presents no special difficulties 
providing the material properties can be well defined. 


943 


8 It is recommended that further tests be conducted similar to 
those performed in this program but with greater emphasis on 
testing panels having dimensional variations in order to develop 
design criteria and life prediction techniques. Such testing 
should be performed on those structural materials and design 
concepts that emerge as the major candidates for flight vehicle 
applications as materials development and manufacturing 
techniques progress. 

REFERENCES 


1 R D. Blevins and I. Holehouse, "Thermo-Vibro Acoustic Loads and 
rkigue of Hypersonic Flight Vehicle Structure," Rohr, Inc. 
Engineering Report RHR 96-008, February 1996. 

2. United States Air Force Systems Command, Flight Dynamics 
Laboratory Final Technical Report, Contract No. F33615-87-C-33^^/, 
to be published. 

3. R. D. Blevins, "Fatigue Testing of Carbon-Carbon Acoustic Shaker 
Table Test Coupons," Rohr, Inc. Engineering Report RHR 91-087, 
September 1991. 

4. R. D. Blevins, "An Approximate Method for Sonic Fatigue Analysis 
of Plates and Shells," Journal of Sound and Vibration, Vol. 129, 
51-71, 1989. 

5. J. W. Miles, "On Structural Fatigue Under Random Loading," 

Journal of Aeronautical Sciences, Vol. 21, November 1954. 


944 


A. Test Configuration for 
Material Coupons 


Fixture 


Specimen 
■ O 



B. Test Configuration for 
Carbon-Carbon Integral 
Stiffener Specimens 


Accelerometer 



t 


C. Test Configuration For 
■ Carbon-Carbon Mechanically 
Fastened Stiffener and All 
Titanium Diffusion Bonded 
Joint Specimens 



FIGURE 1 Typical Strain Gauge Locations and Test Configurations 
for Material Coupon and Joint Subelement Shaker Test 
Specimens 


945 






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CD 

CD 

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FIGURE? Finite Element Solution for In-Phase Overall Mode — 
Carbon-Carbon Panel, Concept 3 


950 




STRRIN [Ins xlB^G/Inl 






952 


SCALING CONCEPTS IN RANDOM ACOUSTIC FATIGUE 

BY 

Marty Ferman* and Howard Wolfe** 

ABSTRACT 

Concepts are given for scaling acoustic fatigue predictions for application to 
extreme environmental levels based on testing “ scaled” structures at 
existing, lesser environmental levels. This approach is based on scaling a test 
structure to fit within the capabilities of an existing test facility to attain 
fatigue results, and then using analytical extrapolation methods for predicting 
the full scale case to achieve accurate design results. The basic idea is to 
utilize an existing acoustic fatigue facility to test a structure which has been 
designed (scaled) to fatigue within that facility’s limits, employing the 
appropriate structural properties (such as thinning the skins, etc.). Then, the 
fatigue life of the actual structure is determined by analytically scaling the test 
results to apply to the full scale case ( thicker) at higher noise levels for 
example. Examples are given to illustrate the approach with limits suggested, 
and with the recognition that more work is needed to broaden the idea. 

BACKGROUND 

While it is important to continually expand the capability of acoustic test 
facilities , it is perhaps equally important to be able to work with existing 
facilities at any time. That is, facility expansions, enhancements , and 
modernization’s should always be sought from time to time, so long as 
practical and affordable from cost effective considerations. Limits should be 
pushed to accommodate larger sizes of test specimens with higher noise levels 
with wider ranges of frequencies, with wider ranges of temperatures, and with 
better capabilities for applying pressures along with any one of several types 
of preloads. These are costly considerations and require considerable time to 
accomplish. Facility rental can be used in some cases to bolster one’s testing 
facilities, however if the application suggests a situation beyond any available 
facility for the required design proof, then an alternate is needed. Thus the 
scaling concept suggested here is a viable and useable possibility. 

The Author’s basis for the approach stems from their extensive, collective, 
experience in Structural Dynamics, especially work in Acoustic Fatigue, 
Fluid-Structure Interaction, Buffet, and Aeroelasticity/Flutter, and 
particularly from experience with flutter model testing, in which it is quite 
common to ratio test results from a model size to full scale for valid 


Assoc. Prof, Aerospace and Mech. Engr. Dept, Parks College, St. Louis 
Univ., Cahokia, IL, 62206, USA 

** Aerospace Engineer, Wright Laboratory, Wright Patterson AFB, OH, 
45433, USA 


953 




predictions . Flutter is a well recognized area where model data is commonly 
used in nondimensional form to establish design margins of safety, as typified 
in Ref (1-2). Flutter can be nondimensionalized quite broadly as pointed out 
in many works, and is clearly done for a wide range of general cases using the 
“so-called” Simplified Flutter Concept, Ref (3). The degree of the use of the 
flutter model scaling rules varies considerably today, because some people 
are testing as much or more than ever, while others are testing less and relying 
more heavily on advanced theories such as Computational Fluid Dynamics, 
CFD. However, the basic ideas in flutter model scaling are still POWERFUL! 
In fact, this concept has fueled the Author’s desires to develop the 
“acoustical scaling” used in the approach presented. Moreover, when starting 
to write this paper, the Authors realized that this type of scaling is also 
common to many related areas of structural dynamics, and thus chose to 
include some examples of those areas to emphasize the main point here! 

For example, experience in fluid-structure interaction and fatigue of fuel tank 
skins, a related work area, serves as another example of scaling structures to 
demonstrate accurate predictions with widely varied environmental levels, 
and a multitude of configurations. Scaling and nondimensional results were 
used extensively in Ref (4 -13), and are cited here because of the immense 
data base accumulated. The work at that time did not necessarily define 
scaling as used here, but hindsight now suggests that there is a clear relation. 

It is becoming well recognized that Buffet is easily scaled , and many 
engineers and investigators are now employing scaling of pressures from 
model to full size applications, and are also using scaled model response to 
predict full scale cases . Some of the earliest and some of the more modern 
results clearly show this aspect. For example. Ref (14 -16) are typical, quite 
convincing, and pace setting regarding scaled data. Buffet models which are 
much more frail that the full-scale cases are used to develop data for full scale 
applications, and besides giving full sized results, provide a guide to safe 
flight testing as has been done more extensively with flutter testing. 

Obviously, acoustical response and fatigue phenomenon are also 
nondimensionalizable and scaleable. Ref (17-18), for example. This point is 
being taken further here; that is , scaling will be used to take better advantage 
of limited facility testing capability to predict more severe situations, as is 
used in the case of flutter model testing where a larger specimen is predicted 
from tests of a smaller structure using similarity rules. Here in the acoustic 
application, a thinner , or otherwise more responsive specimen, is tested and 
then analytical means are used to make the prediction for the nominal case. 

APPROACH 

The method is shown here is basically an extension of the flutter model 
scaling idea, as applied to acoustical fatigue testing with a particular emphasis 
on random applications. The technique will also work for sine type testing in 


954 


acoustical fatigue, and perhaps it will be even more accurate there, but most of 
today’s applications are with random testing, notably in the aircraft field. 
Thus it is in this area where the method should find more application . The 
Authors have a combined professional work experience of some 70+ years 
and thus have tried to focus this extensive background on an area where gains 
can be made to help reduce some costs while making successful designs, by 
using lesser testing capability than might be more ideally used. It is believed 
that the best testing for random acoustic fatigue, is of course, with (a) the 
most highly representative structure, and as large a piece as can be tested, 
both practically and economically, (b) the most representative environmental 
levels in both spectrum shape and frequency content, (c) test times to 
represent true or scaled time, as commonly accepted, (d) temperatures should 
be applied both statically and dynamically, and finally (e) preloading from 
pressures, vibration, and from boundary loading of adjacent structure. 
Frequently, testing is done to accomplish some goal using a portion of these 
factors, and the remainder is estimated . Thus the Authors believed that there 
is a high potential to extend the flutter model approach to acoustical 
applications. 

Recall that in the flutter model approach , the full scale flutter speed is 
predicted by the rule 


((Vf)a)p- [ ((Vi.)m)e/ (( Vf)m)c] X [ ((Vf)a)c] (^) 

where Vp is flutter speed, the subscripts M and A refer to model and aircraft 
respectively, the subscript C refers to calculated, and the subscript P refers to 
predicted. Thus the equation suggests that the full scale predicted flutter speed 
is obtained by taking the ratio of experimental to calculated flutter speed for 
the model and then multiplying by a calculated speed for the airplane. These 
flutter model scaling ideas are covered in any number of References, i.e. Ref 
(lo), for example. 

The same concept can be utilized in acoustic fatigue, i.e. the strain at fatigue 
failure relation, (8,N) can be scaled from model structure tested at one level 
and then adjusted for structural sizing and environmental levels. This relation 
can be addressed as done for the flutter case: 

((e,N)a)p=[(( s,N)m)e / ((e,N)m)c] X [((e.N)a)c] (2) 

where s is strain, and N is the number of cycles at failure, where as above in 
Eq (1), the subscripts M and A refer respectively to Model and Full Scale for 
parallelism, while the subscripts E, C, and P have the same connotation again, 
namely, experimental, calculated, and predicted. Thus the full scale case is 


955 


predicted from a subscale case by using the ratio of experimental to 
theoretical model results as adjusted by a full scale calculation. Flutter model 
scaling depends upon matching several nondimensional parameters to allow 
the scaling steps to be valid. While these same parameters are, of course, not 
necessarily valid for the acoustic relationships, other parameters unique to 
this acoustical application must be considered, and will be discussed. 
Accurate predictions for the method relies on extensive experience with the 
topic of Acoustic Fatigue in general, because concern is usually directed 
towards the thinner structure such as; panels, panels and stiffeners, and panels 
and frames, bays (a group of panels), or other sub-structure supporting the 
panels. These structures are difficult to predict and are quite sensitive to edge 
conditions, fastening methods, damping, combination of static and dynamic 
loading, and temperature effects. Panel response prediction is difficult, and 
the fatigue properties of the basic material in the presence of these complex 
loadings is difficult. However, the experienced Acoustic Fatigue Engineer is 
aware of the limits, and nonnally accounts for these concerns. Thus the 
method here will show that these same concerns can be accounted for with the 
scaling approach through careful considerations. 

The Authors believe that the method is best explained by reviewing the 
standard approach to acoustic fatigue, especially when facility limits are of 
major concern. Fig (1) was prepared to illustrate these points of that 
approach. Here it is seen that key panels for detail design are selected from a 
configuration where the combination of the largest, thinnest, and most 
severely loaded panels at the worst temperature extremes and exposure times 
are considered. These can be selected by many means ranging from empirical 
methods, computational means, and the various Government guides. Ref (17- 
18), for example. Then detailed vibration studies are run using Finite 
elements , Rayleigh methods. Finite Difference methods, etc. to determine the 
modal frequencies and shapes, and frequently linearity is assessed. Then 
acoustical strain response of the structure is determined for sine, narrowband, 
and broadband random input to assess fatigue life based on environmental 
exposure times in an aircraft lifetime of usage. These theoretical studies are 
then followed by tests of the worst cases, where vibration tests are conducted 
to verify modal frequencies, shapes, and damping, and linearity is checked 
again for the principal modes. This is followed by acoustical strain response 
tests where the strain growth versus noise levels is checked, again employing 
sine, narrowband and broadband random excitation. Note the figure suggests 
that data from the vibration tests are fed back to the theoretical arena where 
measured data are used to update studies and to correlate with predictions, 
especially the effect of damping on response and fatigue, and of course, the 
representation of nonlinearity. Also, the measured strain response is again 
used to update fatigue predictions. These updates to theory are made before 
the fatigue tests are run to insure that nothing is missed. However, in this 


956 


case, the required sound presssure level SPL in (dB) is assumed to exceed the 
test chamber’s capability. Thus , as shown in the sketch in Fig (2) the key 
strain response curve, s vs dB, is extrapolated to the required dB level. This 
data is merged with the strain-to-failure curve at the right to establish the cycle 
count, N, giving the (s, N) point for this case. The extrapolated data provides 
some measure of the estimated life, but again is heavily dependent upon the 
accuracy of the basic strain response curve, and is especially dependent on 
whether high confidence exists at the higher strains. Linear theory is also 
shown in this case, indicating it overpredicts the test strain response and hence 
shows a shortened fatigue life compared to test data, as is generally the case 
in today’s extreme noise levels. This illustration is highly simplified, 
because experienced designers readily know that it is difficult to predict even 
simple panels accurately at all times, let alone complex and built-up structure 
consisting of bays (multi-panel); this will addressed again later in the paper. 

The new concept of scaled acoustic fatigue structures is shown on the sketch 
of Fig (3) where the standard method is again shown, but smaller in size, to 
refocus the thrust of the new idea. The scaling process parallels the standard 
approach, and actually complements that approach, so that the two can be run 
simultaneously to save time, costs, and manpower. Here the panel selection 
process recognizes that the design application requires environments far in 
excess of available facility capability. Thus the scaling is invoked in the 
beginning of the design cycle. As the nominal panels (bays) are selected and 
analyzed for vibration, response and fatigue, scaled structures are defined to 
provide better response within the existing chamber ranges so that they can be 
fatigued and then the results can be rescaled to the nominal case. In this 
manner, appropriate designs can be established to meet safety margins with 
more confidence, and will avoid costly redesign and retrofitting at 
downstream stages where added costs can occur and where down times are 
difficult to tolerate. The concept is further illustrated in the sketch of Fig (4). 
Here the strain response curve of the nominal case and that of the scaled 
version are combined with strain to failure data ( coupon tests) to show fatigue 
results. Note the strain response for the nominal case at the highest dB level 
available gives the fatigue value at point A, while the extrapolated data for 
this curve gives point B. The scaled model being more responsive gives the 
point C, and when rescaled gives the point D which differs slightly from the 
extrapolated point B as it most likely will, realistically. More faith should be 
placed on data from an actual fatigue point than a point based on the projected 
strain response curve. Note, Fig (5) illustrates the winning virtue of the 
scaleable design. The figure shows a hypothetical set of test data for the (s ,N) 
for a structure for various SPLs for the nominal case, open circles, and for the 
fatigue results of the scaled model , closed squares. The scaled model was 
assumed to be thinner here for example, and that the scaled data is also 


957 


rescaled to fit the nominal curve here. The most interesting aspect is shown 
by the two clusters of data, denoted as A and B where there are rough circles 
about drawn about the clusters. Here the emphasis is that tests of the scaled 
model ( and rescaled data) are used to find the higher strain conditions which 
cannot be found from the nominal case. In both cases at the highest strain 
levels, the facility is used to its limits, but with enough testing with the thinner 
case, adequate data is available to make the prediction more accurate using 
Eq (2) for the final correlation as shown here. The statistical scattering of the 
scaled data will be an accurate measure for the nominal case, particularly 
when compared to estimates based on extrapolation of the strain response for 
the nominal case. There are many cautions to be noted with this approach as 
there are with all acoustic fatigue methods, and of course, tests. First, the 
linearity of the modes, either in unimodal sine excitation, multi-mode sine, 
narrow band or broadband random must be carefully handled. The strain 
response of individual locations throughout the structure must be carefully 
monitored in calculations and tests so that strain response is truly understood 
and used to define fatigue life carefully. This is difficult to do in many 
applications where widely varying conditions and durations require some type 
of Miner Rule combination to provide a true measure of fatigue. Similarly, 
strain risers at fasteners, discontinuities, holes, frames, stiffeners, material 
changes along with temperature gradients, temperature transients, require 
final “tweeks” to predictions, regardless. Nonlinearity, especially in the 
multi -mode case, is one of the most formidable foes to conquer for any 
application. 


APPLICATIONS AND EXAMPLES 

The tests of an Aluminum panel of size 10x20 in. and with a thickness of 
0.063 thickness , Ref (19), will be used to illustrate the technique. The panel 
has approximately fixed-fixed edge conditions, and is quite nonlinear in 
terms of strain response. Ref (19). The measured strain response for the panel 
is shown in Fig (6) along with an estimated response curve for a thinner panel 
( 0.040 in ) based on the test data. For this case it is assumed that data were 
needed at 175 dB, while the facility could only achieve 164 dB. The strain 
response for the thinner case was estimated using the classical equation for 
the amplitude of response, 5 ; 


\\fdxdy\pSD,{f)] 


(3) 


where M is the generalized mass, co is the natural frequency, § is the viscous 
damping factor , (j) is mode shape, PSDp is the pressure Power Spectral 
Density, and x,y are the positional coordinates along the plate. Since strain , 
s, is proportional to the amplitude. 


958 



s= (t/2)(3(ti^/ax^)S 


(4) 

Combining Eq (3 and 4) shows that the strain response curves are 
proportional to the thickness factor, as given by: 

s 2 ~ ( ti / t2) (5) 

However, it must be noted that this case is nonlinear, and thus, this result is 
not exactly correct, but simply used for an illustration here. The actual data for 
the 0.063 thickness is extrapolated to a required 175 dB, showing a strain of 
1000 micro in/in. The estimated curve for the thickness of 0.040 in. shows ,of 
course, a greater response at all dB levels as it should, and moreover shows 
that only 150 dB are needed to achieve the 1000 micro - strain condition. 
Moreover, the thinner panel will exhibit large enough strains at the lower 
SPLs to improve the fatigue curve where the thicker panel is insensitive. 
Taking the example a step further, the fatigue point of the nominal case is 
shov^Ti on a strain to failure plot in Fig (7), employing beam coupon tests of 
Ref (11), which were shown to be excellent correlators with panel fatigue in 
the collection of work in Ref (6-13). The fatigue point for the 164 dB 
excitation, 800 micro-strain, is shown as a triangle, while the extrapolated 
data for 175 dB is shown as the flagged triangle. One test point exists for the 
nominal case. Ref (20), and is shown by a star symbol. Data for the thinner 
panel are shown as circles at the various strain to cycle count cases for the 
various SPLs corresponding to the beam curve. Notably, these points can be 
seen to produce shorter fatigue cases as they should due to increased strains, 
but note that they are also at lower frequencies which would give a longer test 
time than if they were the nominal thickness. The scaled model is seen to 
produce the same point as the extrapolated case in this hypothetical case for 
the 1000 microstrain case ( again, at two different dB levels for two 
thicknesses). A SPL of 150 dB, rather benign, is seen to be quite effective. 
The actual fatigue point at 164 dB for the nominal case required 3 hours and 
was predicted to be 2.8 hrs. The estimated fatigue for the extrapolated case of 
175 dB was estimated to be 1.7 hrs, while the scaled point from the thinner 
panel was estimated to be 2.2 hours which is slightly off, but the Authors 
have had to rely on log plots for much of the data and thus lack someaccuracy. 
Because of lack of actual data, the scatter from the estimate vs the test of the 
nominal case was used to scatter the estimate for the 1000 microstrain case, 
flagged dark circle, as if the use of Eq (2) had been employed directly. One 
must be careful here, because there can be a vast difference between theory 
and test, and this can mislead inexperienced persons applying these methods. 
As noted earlier, related work in fluid-structure and buffet , actually 
demonstrate this type of scaling. To illustrate, several figures are republished 


959 


here to make this point rather clear. Fig (8) of Ref (8 ) shows the dynamic 
bending strains in the bottom panel of an otherwise rigid fuel tank which is 
being excited vertically with moving base input. The vertical axis is strain 
while the abscissa is the number of g’s input. Three panel thicknesses and four 
depths of fluid (water in this case) were used. Note the sharp nonlinear effect 
in the response, rather than linear response growth as force increases. 
Interestingly, the data was nondimensionalized into the curve of Fig (9), Ref 
(8), which was originally intended for a design chart to aid in developing 
strain response characteristics for use in fatigue. This curve displays a 
parameter of response as the ordinate vs an excitation parameter on the 
abscissa. Here, E is Young’s modulus, p is density, t is thickness, a is the 
panel length of the short side, h is fluid depth, and the subscripts, p and F refer 
to panel and fluid, respectively. A point not realized previously is that the 
scaling shows that the thinner case can be used to represent the thicker panel 
under the appropriate conditions and when nonlinearity is carefully 
considered. More data with the thinner panels at the extreme conditions were 
unfortunately not taken in several cases of strain response because of concern 
with accumulating too many cycles before running the actual fatigue tests; 
else the thinner cases could have shown even more dramatically the scale 
effect. 

Buffet has been of more interest in the past 15 years because of high angle of 
attack operation of several modem USAF fighters. Much effort was placed 
upon research with accurately scaled models to detennine if these could be 
employed as in prior flutter work. The answer was YES! Several figures were 
taken from Ref (16) to illustrate scaling of data from a model of, a fraction 
of the size of a fighter, to the full scale quantity. Fig (10) shows the correlation 
between scaled-up model data, flight test, and two sets of calculations over a 
wide range of aircraft angle of attack for the F/A-18 stabilator. The data is for 
inboard bending and torsion moment coefficients produced by buffeting 
loads. The scaled model data correlates well, the calculations using Doublet 
Lattice (DLM) aerodynamics is close, while the strip theory is not as accurate. 
Fig (11) shows similar type of data for the F/A-18 Vertical Tail for outboard 
bending moment coefficients. Here a wider range of angle of attack was 
considered, and again scaled model data and calculations are close to aircraft 
values. Both cases suggest that model data can be used to supplement full 
scale work and that when combined with theory , are a powerful aid to full 
scale analysis and tests. These tests can be used early in the aircraft design 
cycle to insure full scale success. 

CONCLUSIONS AND RECOMMENDATIONS 
An attempt was made to employ a view of acoustical scaling different from 
that usually taken. The idea is to develop data for a model that fits within a 
test facility’s capability and then by using analytical methods, adjust these 
results to the nominal case using factors from the test based on the ratio of 


960 


experimental to calculated data. This is analogous to the flutter model 
approach. One example is offered, and similar results from related scaling in 
fluid -structure and buffet work were shown to further the point. While more 
work is needed to fully display the concept, enough has been done to inspire 
others to dig-in and more fully evaluate the approach. The Writers intend to 
do more research, since they fully appreciate this difficult task. 

REFERENCES 

1. Scanlon , R.H., and Rosenbaum, R., “ Introduction to The Study of Aircraft 
Vibration and Flutter”, The MacMillian Company, New York, 1951 

2. Bisplinghoff, R.L., Ashley , H. and , Halfman, R. L., “Aeroelasticity”, 
Addision-Wesley Publishing Co., NY, Nov. 1955, pp. 695-787 

3. Ferman, M. A., “Conceptual Flutter Analysis Techniques - Final Report” 
Navy BuWeps Contract NO w 64-0298-c, McDonnell Report F322, 10 Feb. 
1967 

4. Ferman, M.A. and Unger, W. H., “Fluid-Structure Interaction Dynamics in 
Fuel Cells”, 17th Aerospace Sciences Meeting, New Orleans, La. Jan 1979 

5. Ferman, M.A. and Unger, W. H. , “Fluid-Structure Interaction Dynamics in 
Aircraft Fuel Cells”, AlAA Journal of Aircraft, Dec. 1979 

6. Ferman, M.A. , et al , “ Fuel Tank Durability with Fluid-Structure 
Interaction Dynamics ,” USAF AFWAL TR-83-3066, Sept. 1982 

7. Ferman, M..A., Unger, W. H., Saff, C.R., and Richardson, M.D., “A New 
Approach to Durability Predictions For Fuel Tank Skins” , 26th SDM, 
Orlando, FL, 15-17 April 1985 

8. Ferman, M. A. , Unger, W. H., Saff, C.R., and Richardson, M.D. , “ A 
New Approach to Durability Prediction For Fuel Tank Skins”, Journal of 
Aircraft, Vol 23, No. 5, May 1986 

9. Saff, C.R., and Ferman, M.A, “Fatigue Life Analysis of Fuel Tank Skins 
Under Combined Loads”, ASTM Symposium of Fracture Mechanics , 
Charleston, SC, 21 March 1985 

10. Ferman, M.A., Healey, M.D., Unger, W.H., and Richardson, M.D., 
“Durability Prediction of Parallel Fuel Tank Skins with Fluid-Structure 
Interaction Dynamics”, 27th SDM, San Antonio, TX, 19-21 May 1986 

11. Ferman, M.A, and Healey, M.D., “Analysis of Fuel Tank Dynamics for 
Complex Configurations, AFWAL TR -87-3066, Wright-Patterson AFB, OH, 
Nov 1987 

12. Ferman, M.A., Healey, M.D. and Richardson, M.D.,” Durability Prediction 
of Complex Panels With Fluid-Structure Interaction”, 29th SDM, 
Williamsburg, VA, 18-20 April 1988 

13. Ferman, M.A., Healey, M.D., and Richardson, M.D., “A Dynamicisf s 
View of Fuel Tank Skin Durability, AGARD/NATO 68th SMP, Ottawa, 
Canada, 23-28 April 1989 

14. Zimmerman, N.H. and Ferman, M.A., “Prediction of Tail Buffet Loads 
for Design Applications, USN Report, NADC 88043-30, July 1987 


961 


15. Zimmerman, N.H., Ferman, M.A., Yurkovich, R.N, “Prediction of Tail 
Buffet Loads For Design Applications”, 30th SDM, Mobil, AL , 3-5 April 
1989 

16. Ferman, M.A., Patel, S.R., Zimmemian, N.H., and Gerstemkom, G., “ A 
Unified Approach To Buffet Response Response of Fighters”,AGARD/NATO 
70th SMP, Sorrento, Italy, 2-4 April 1990 

17. Cote, M.J. et al, “Structural Design for Acoustic Fatigue”, USAF ASD- 
TR-63-820, Oct 1963 

18. Rudder, F.F., and Plumblee, H.E., Sonic Fatigue Guide for Military 
Aircraft” AFFDL-TR-74-112, Wright-Patterson AFB, OH, May 1975 

19. Jacobs, J.H., and Ferman, M.A. , Acoustic Fatigue Characteristics of 
Advanced Materials and Structures, “ AGARD/NATO SMP , Lillehammer, 
Norway, 4-6 May 1994 

20. McDonnell Douglas Lab Report, Tech. Memo 253.4415, Acoustic Fatigue 
Tests of Four Aluminum Panels, Two With Polyurethene Sprayon”, 27 June 
1984 



Figure 1 - Standard Acoustic Fatigue Design Method 


962 









MICROSTRAIN - 


HYPOTHETICAL EXAMPLE 



SOUND PRESSURE LEVEL -dB N-CYCLES TO FAILURE 


Figure 2 - General Method for Estimating Fatigue Life at SPL 
above Test Facility Capability 





Figure 3 - Scaling Method Fits-in with General Design 
Cycle for Acoustic Fatigue 


963 







SOUND PRESSURE LEVEL - dB 


N@ cIBreq , Nom. Ext. 
N @ dBj^, Re-scaled 
N @ dBj^, Scaled Model 
N - CYCLES TO FAILURE 


Figure 4 - Hypothetical Examples of Acoustic Scaling 
to Tests at Higher SPLs 


HYPOTHETICAL CASE 



Figure 5 - Statistical Aspects of Scaling 





MICROSTRAIN - 


^ MDC Tests 
Ref. (19) 



Figure 6 - Strain Response of an Aluminum Panel 

(10 X 20 X 0.063 in. 7075 T6) Narrowband Random 



Figure 7 - Example of scaled Model of the 10 x 20 in. 
Aluminum Panel 


965 



Symbol Panel Fluid 
Thickness Depth 

(in.) (in.) 

□ 

0.032 

11.0 

0 

0.040 

11.0 

o 

0.063 

11.0 

0 

0.032 

8.0 

A 

0.040 

8.0 

o 

0.063 

8.0 

0 

0.032 

4.0 


0.040 

4.0 

• 

0.063 

4.0 


Figure 8 - Dynamic Strain vs Excitation Level - Sine 



Figure 9 - Dynamic Strain Parameter vs Input Parameter 

Mean Strain at Fluid Depth 


966 



0.04 


Inboard Bonding 


Inboard Torsion 


0.032 

RMS Buffet 0.024. 

Moment 
Coeffidant, 

0.01 e 

0.008 

0 . 

0 4 8 12 16 20 24 

Angle of Attack - degrees 






A 






C 

1 




1 

r 

1 0 

< 

) 














• 

0 

_^ 

— 

Right t» 
Scaladt 
Calculal 
italculat 

— 

St 

ipmods 

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theory) 

) 

— 

A 

L 





i 

> 

(j 

I o 




( 

0 

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0 4 8 12 16 20 24 


Angie of Attack - degrees 


Figure 10 - F/A-18 Stabilator Buffet Correlation Study 


0.048 

0.04 

0.032 

RMS 
Moment 

CoefficiantO.024 
BM/(QL3) 

0.016 

0.008. 

0 

16 24 32 40 48 56 

Angle of Attack - degrees 

Figure 11 - F-18 Vertical Tail Buffet Response Moment Coefficients for 
Angle-of-attack Variations 


Outboard Bending Moment (70% Span, 45% Chord) 

j—I j j 


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Zi 

1 










967 







968 


ACOUSTIC FATIGUE II 



THE DEVELOPMENT AND EVALUATION OF A NEW 
MULTIMODAL ACOUSTIC FATIGUE DAMAGE MODEL 


Howard R Wolfe 
WL/FIBG Bidg24C 
2145 Fifth St Ste2 
Astronautics 

Wright-Patterson AFB, OH 
45433-7006 
USA 


Robert G. White 
Head of Department 
Department of Aeronautics and 

University of Southampton 
Southampton, S017IBJ 
UK 


ABSTRACT 

A multimodal fatigue model has been developed for flat beams and plates. 
The model was compared with experimental bending resonant fatigue 
lifetimes under random loading. The method was accurate in predicting 
cantilevered beam fatigue lifetimes, but under predicted clamped- clamped 
beam test results. For the clamped plate tests, one calculation was accurate 
and the other predicted about half the test lifetimes. The comparisons and 
the parameters affecting them are presented. 

INTRODUCTION 

While the single mode acoustic fatigue theory is satisfactory for sound 
pressure levels around 158 dB overall and below, there is evidence in the 
literature [1, 2, and 3] that above this level the accuracy of the simple 
response prediction method decreases with increasing sound pressure 
levels. The purpose of the paper is two fold, first to develop a multimodal 
acoustic fatigue life prediction model ^d secondly, to evaluate its accuracy 
in estimating the fatigue life theoretically by comparing predictions with 
experimental results. 

FATIGUE MODEL DEVELOPMENT 

Many fatigue models are found in the literature. The Miner single mode 
model used by Bennouna and White [4] and Rudder and Plumblee [5] was 
selected to develop a multimodal nonlinear model. The fundamental 
formulation is given by, 

N.-fs^r «) 


969 


where N,. is the total number of cycles to failure, P(e) is the peak strain 
probability density. N is the total number of cycles to failure at incremental 
constant amplitude strain levels derived from a sinusoidal strain versus 
cycles to failure curve. To calculate the fatigue life in hour, Eq (1) can be 
expressed as, 

t (hours)= 



where t is time, Pp (sd) is the peak standard deviation probability density, 
Nc is the total number of cycles to failure at a specified strain level and ^ is 
the cyclic frequency. When the mean value is not zero, which is the case 
with axial strain in the beam or plate, the rms value is not the standard 
deviation. The standard deviation is usually employed to compute the time 
to failure. Most of the S-N curves or e -N curves are approximated as a 
straight line on a logarithmic graph. The relationship between the surface 
strain and the cycles to failure is then. 


8 = 



(3) 


where K is a constant and a the slope of a straight line on a log-log graph. 
The cyclic frequency 4 for a single mode case is taken as the frequency of 
the associated resonance. Two types of peak probability density techniques 
were investigated from multimodal nonlinear strain responses [3]. These 
were called major peaks and minor peaks. The major peaks were counted 
for the largest peaks between zero crossings. The minor peaks were 
counted for all stress reversals or a positive slope in the time history 
followed by any negative slope. The effective cyclic frequency is much 
lower for the major peak count than the minor peak count. However, the 
peak probability density functions or PPDFs compared for these two cases 
were almost the same. The major peak method was selected for further 
study. 

Given a particular peak probability density curve from a measured 
response in an experiment, the number of peaks and the sampling time tg 
can be used to determine the effective multimodal cyclic frequency, 

fcm = number of peaks / tg (4) 


970 



where is the effective multimodal cyclic frequency. Substituting the 
multimodal cyclic frequency into Eq (4), 


t (hours) = 



Pp(sd) 

K/(e)‘/“ 


3600xfcm 



(5) 


This model accounts for the effects of axial strains which cause the mean 
value not to be zero, nonlinear response and multimodal effects. If the 
mean value is zero, then the standard deviation is equal to the rms value 
and Eq (5) reduces to Eq (2). 

FATIGUE MODEL COMPARISON WITH EXPERIMENTAL 

RESULTS 


The peak probability density function (PPDF) is needed or preferably 
the time history from strain or dynamic response measurements to 
evaluate the fatigue model developed. Also needed are sinusoidal 8 -N 
curves for the structure, and knowledge of its boundary conditions and the 
equivalent multimodal cyclic frequency. 

Two types of peak probability density techniques were investigated from 
multimodal nonlinear strain responses [3]. These were called major peaks 
and minor peaks. The major peaks were counted for the largest peaks 
between zero crossings. The nunor peaks were counted for all stress 
reversals or a positive slope in the time history followed by any negative 
slope. The effective cyclic frequency is much lower for the major peak coimt 
than the nunor peak coimt However, the peak probability density 
functions or PPDFs compared for these two cases were almost the same. 
The major peak method was selected for further study. 

Comparison with Beam Data: 

The K and a terms were calculated from 8-N and S-N data, where S 
is stress, using Eq (3). Selecting two values of strain and their 
corresponding cycles to failure, yields two simultaneous equations 
which were solved for K and a. Two sinusoidal E-N curves for 
BS1470-NS3 aluminum alloy which has a relatively low tensile strength 
were obtained from Bennouna and White [4 Fig 8]. These were for a 
cantilevered beam and a clamped-clamped (C-C) beam as shown in Fig 
1. The K and a terms calculated were used to compute the cycles to 
failure, Nc, for each strain level. Table I shown is the same as Table I in 
reference [4] except was calculated from Eq (3) to sum the damages. 


971 


Delta is the sample size. The cyclic frequency was for one mode the one 
resonant response frequency. The time to failure in hours using Eq (2) 
for the cantilevered beam was 16.6 hours compared with 16.2 predicted 
theoretically [4], 15.3 and 15.9 obtained experimentally [4]. Both 
theoretical results were essentially equal, but slightly higher than the 
test results. The time to failure from Eq (2) for the (C-C) beam was 3.04 
hours compared with 2.53 theoretically [4] and 5.25 and 5.92 
experimentdly [4]. Both theoretical results were about one half of the 
test results. The K and a terms, the theoretical fatigue life times and the 
experimental fatigue life times are listed in Table H. The table contains 
three sections: lifetimes calculated using a strain gauge PPDF, a 
displacement PPDF and the Gaussian and Rayleigh PDFs. As noted in 
reference [4], failure occurred much earlier for the C-C beam than the 
cantilevered beam for the same strain level. This was attributed to the 
influence of a large axial strain in the clamped- clamped beam. 

Comparison with Plate Data: 

Two fatigue tests were conducted to provide some additional limited 
data for comparison with the fatigue model developed. These tests 
used the base excitation method with a 1.09x10® N (20,000 Ibf) 
electrod)mamic shaker. The clamping fixture consisted of a flat 
aluminum alloy 6061-T6 plate 19 mm thick and four clamping bars of 
equal thickness. The radius of curvature of the clamping edges was 4.76 
mm to prevent early fatigue failure. A four bar clamping arrangement 
was selected to prevent buckling of the plate while torquing the 
clamping bolts. The undamped size was 254 x 203 x 1.30 mm which 
results in a 1.25 aspect ratio. Strain gauges were bonded along the center of 
the larger dimension (SG 2) and at the center of the plate (SG 3). 
Displacements were measured with a scanning laser vibrometer at the 
center of the plate. An accelerometer was moimted on the shaker head to 
determine the acceleration imparted to the damped plate. A flat 
acceleration spectral density was used between 100-1500 Hz. Recordings 
were taken at increasing levels of exdtation up to the fatigue test level. The 
time to detecting the first fatigue crack was recorded for each plate. 

The constants K and a were calculated from random single mode S-N 
data for 7075-T6 aluminum alloy [5 p 489] shown in Fig 1, with 

K = 1.01x10^^ and a = - 0.175. The rms stress (Srms) ^^s changed to rms 
strain, Srms = EBrms, where E is Young's modulus. The stress was 
measured half way between two rivets along the center line between the 
rivets on the test specimen. The strain gauge location, stress 
concentrations, and the boundary conditions greatly affect the strain 


972 


level measured. Correction factors are needed for a different set of 
conditions and to convert random data to sinusoidal data. Sinusoidal 
£ -N bending coupon curves for 7075-T6 aluminum alloy were difficult to 
find. S-N curves were found for an aerospace material with both 
sinusoidal and random excitations. These curves were nearly parallel. 
The sinusoidal strain was 1.38 times larger than the random strain for 

10^ cycles. Multiplying the constant K for the 7075-T6 material by 1.38 
resulted in K = 1.40x10^^. 

Early strain gauge failures prevented strain measurement above 500 
microstrain with 20.7 g rms shaker excitation. The fatigue test level was 
115 g rms and the response contained at least six frequency response 
peaks. The major peak strain PPDFs were determined for 5.32 g rms 
and 20.7 g rms as shown in Fig 2 with the Gaussian PDF. Compared 
with the Gaussian distribution, an increased number of peaks occurred 
greater than 1 sigma and smaller than -1 sigma. Also a larger number of 
peaks occurred around zero. The PPDF determined from the 20.7g rms 
test case was used to predict fatigue life, but a new strain estimate was 
needed since the excitation level increased 5.6 times. The displacement 
is directly related to the strain at each excitation level. Displacement 
measurements at the fatigue test level were used to estimate the strain 
level shown in Fig 3. The estimated strain from the figure was 770 
microstrain for SG 2. The scale of the displacement measurements was 
adjusted to coincide with strain measurements at increasing increments 
of shaker excitation. 


The equivalent cyclic multimodal frequency is needed to predict the 
fatigue life. Prediction of the linear modal frequencies is carried out by 
a variety of methods. Usually the first mode prediction is the most 
accurate. The cyclic multimodal nonlinear frequencies have been 
studied for two clamped beams and two clamped plates [3]. These were 
based upon the peak probability density functions (PPDFs) where the 
peaks were counted for a specific time interval, from which the 
nonlinear cyclic multimodal frequencies were calculated. Generally the 
resonant frequencies increased with increasing excitation levels. Those 
for the two beams increased more rapidly than those for the two plates. 
Very little change was noted for the plates. The equivalent cyclic 
miiltimodal frequency determined via Eq (4) from the SG 2 PPDF was 
348 Hz. 

The time to failure in hours predicted using Eq (5) and the parameter 
mentioned above for the clamped plate was 0.706 hours compared with 
1.17 and 0.92 shown in Table 11. The predicted result was slightly lower 


973 



than the test results. A Srms’^ curve was calculated with the test 
lifetimes available, by determining a new constant K, assuming the 
slope was the same as for the riveted coupon and applying the 
sinusoidal correction factor. The time to failure in hours predicted 
using Eq 5 and the calculated e-N curve for the clamped plate was 
0.274 hours compared with 1.17 and 0.92 shown in Table E. The cyclic 
multimodal frequency used was the same at that determined from SG 2 
and the same strain was used. This prediction was about 1/3 of the test 
results. This method incorporates the failure data at two points. 

The displacement PPDF shown in Fig 4 was used to predict fatigue 
life. The number of displacement peaks increased significantly above 
the strain PPDF around 1 sigma and -1 sigma. The large number of 
peaks around zero was similar to the characteristics observed in the 
strain PPDF. The equivalent cyclic multimodal frequency was 375 Hz, 
slightly higher than that determined from the strain PPDF. However, 
the same frequency (348 Hz) was used to predict fatigue life. The time to 
failure in hours using Eq (5) for the clamped plate was 1.15 hours with 
the riveted e-N curve and 0.446 with the calculated 8-N curve shown 
in Table E. This PPDF improved considerably the prediction. The 
Gaussian and Rayleigh PDFs were used to predict the time to failure 
with the same parameters as those used with the riveted e-N curve. 
The lifetime using the displacement PPDF was 1.15 hours, using the 
Gaussian PDF, 0.600 hours and using the Rayleigh PDF, 0.237 hours, as 
shown in Table E. The Gaussian PDF under predicts by a factor of 2. 
The Rayleigh PDF under predicts by a factor of 5. 

The spread sheets containing PPDF / Ncm data for various sigma 
values were used to determine damage accumulation shown in Fig 5. 
Almost 55% of the normalized damage occurs between -2 and -1 sigma 
and 38% of the damage between 1 and 2 sigma using the displacement 
PPDF. However, the damage is spread more evenly using the available 
strain gauge PPDF. The strain gauge PPDF was recorded at a much 
lower level than the displacement PPDF. The damage accumulation 
compared more closely to Gaussian PDF than the Rayleigh function. 


Damage Model with a Specific Function Describing the PPDF : 

A curve-fitting routine was used to determine a mathematical function 
for a high level strain gauge PPDF for the clamped shaker plate. The 
most important part of the fit is outside the range of -1 to 1 sigma, since 
most of the damage accumulation occurs outside this range. The 
highest ranking function was a tenth order polynomial followed by 


974 


ninth and eighth order polynomial fits. The goodness of fit in order 
from 1 to 14 ranges from 0.9775 to 0.9625, which are very close 
statistically. The tenth order polynonual is, 

y = a 4* bx + cx^ + dx^ + ex"^ + fx^ + gx^ + hx^ + ix^ + jx^ + kx^^ (6) 

where a=0.346, b=-0.0148, c=-0.137, d=-0.054, e=0.090, f=0.043, g=- 
0.0400, h=-0.00976, i= 0.00722, j=6.85xl0"‘, k=<4.36xl0'" . Ranked 
fourteenth is a natural logarithmic function. The function and its 
coefficients are, 

Iny = a + bx + cx^ + dx^ +ex^ 4-fx^ (7) 


where a=-1.088, b=-0.1191, c=-0.1.302, d=0.0104, e=-0.0653, and f=0.0079. 
The function fits the test data similar to the tenth order polynomial and 
may be easier to use. Ranked forty-first is a Gaussian function. The 
function and its coefficients are. 


y = a + b exp|o.5[(x - c) / d]^ | 


(8) 


where a=-0.0968, b=0.4485, c=-0.050 and d=1.45. The function fits better 
for sigma values of 2 or greater than those of -2 sigma and greater. A 
constant coefficient is used to fit the Gaussian function to permit shifting 
the function to fit the test data. This equation can be used in the PPDF 
in Eq (5), 

a + b exp [-0.5[(x - c) / d]^ } 

- [K/(e)f“ 




3600£ 


■cm 


(9) 


where a=-0.0968, b=0.4485, c=-0.050 and d=1.45. 


CONCLUSIONS 

The prediction of multimodal fatigue life is primarily dependent upon 
the peak probability density function (PPDF) which changes shape with 
increasing excitation levels. The next in order of importance is the 
sinusoidal e -N bending fatigue curve and finally the effective multimodal 
cydic frequency. 

A multimodal fatigue model was developed with the PPDF estimated 
from a form of the Gaussian function being useful especially in the 


975 


range of cydes to failure from 10^-10^. The lifetime predication 
calailations for the damped-clamed beam was about one half the 
experimental value. For the plate, the calculations was about one half 
the experimental value. Using riveted coupon fatigue data, the 
calculation was accurate. 


REFERENCES 

1. B.L. Clarkson, April 1994, '‘Review of sonic fatigue technolog/', 
NASA contractor report 4587, NASA Langley Research Center, 
Hampton, Virginia. 

2. R.G. White, October 1978, "A comparison of some statistical 
properties of the responses of aluminium alloy and CFRP plates to 
acoustic excitation". Composites 9(4), 125-258. 

3. H.F. Wolfe, October 1995, "An experimental investigation of 
nonlinear behaviour of beams and plates excited to high levels of 
dynamic response", PhD Thesis, University of Southampton. 

4. M. M. Bennouna, and R. G. White, 1984, "The effects of large 
vibration amplitudes on the dynamic strain response of a clamped- 
clamped beam with consideration on fatigue life". Journal of Sound and 
Vibration, 96 (3), 281-308. 

5. J. R. Ballentine, F. F. Rudder, J. T. Mathis and H.E. Plumblee, 

1968, 'Refinement of sonic fatigue structural design criteria", AFFDL TR 
67-156, AD831118, Wright-Patterson AFB, Ohio. 


TABLE I 

FATIGUE CALCULATIONS USING EQUATION 5.9 

CANTILEVERED BEAM BS 1470-NS3 (REF 4 FIG 8) 


e/sd 

sd 

P£ 

e 

ll£ 

Nc=(2.172 

xl0“/sd)“““ 

PPDF X A 

PPDF X A/Nc 

t 

hours 

0.5 

H 

213 

15079045.41 

n 

1.41256E-07 


1 

|g 

425 

4390936.857 

mam 

7.06000E-07 


1.5 

425 

638 

2125446.064 

WBrn 

1.12917E-06 


2 

425 

850 

1273261.536 

0.138 

1.08383E-06 


2.5 

425 

1063 

854026.6252 

0.060 

7.02554E-07 


3 

425 

1275 

617189.6449 

0.018 

2.91645E-07 


3.5 

TOTAL 

425 

1488 

468372.0662 

0.005 

0.984 

1.06753E-07 

4.16121E-06 

16.7 


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TABLE n 

SUMMARY OF FATIGUE CALCULATIONS 





















Figure 1 8 -N curves for aluminum alloys tested. 



Figure 2 Normalized strain PPDF comparison with a Gaussian PDF. 


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NORMALIZED DISPLACEMENT 
MOJORPPDF 


2 




Figure 4 Normalized displacement PPDF comparisons, 


9 













980 











ACOUSTIC FATIGUE AND DAMPING TECHNOLOGY IN 
COMPOSITE MATERIALS 


By B. Benchekchou and R.G. White 


Abstract 

Considerable interest is being shown in the use of composite materials in 
aerospace structures. Important areas include development of a stiff, lightweight 
composite material with a highly damped, high temperature polymer matrix 
material. The study described in this paper concerns the application of such 
material in the form used in thin skin panels of aircraft and investigation of its 
fatigue properties at room and high temperature. For this purpose, flexural fatigue 
tests have been carried out at two different temperatures and harmonic 
three-dimensional FE analyses were performed in order to understand the 
dynamic behaviour of plates. Random acoustic excitation tests using a 
progressive wave tube, up to an overall sound pressure level of 162 dB, at room 
temperature and high temperatures were also performed in order to investigate the 
dynamic behaviour of panels made of the materials. Various methods for 
including damping in the structure were examined when parameter studies were 
carried out, and conclusions have been drawn concerning optimal incorporation 
of a highly damped matrix material into a high performance structure. 

1-Introduction 

Significant areas of primary and secondary structures in military aircraft operate 
at high temperature and are subjected to high levels of random acoustic loading, 
because of their closeness to jet effluxes. There is then a need to develop a carbon 
fibre reinforced plastic material with a high temperature polymer matrix and high 
fatigue resistance. Highly damped composite structures should be developed in 
order to better resist dynamic loading and to have an enhanced fatigue life. Work 
previously carried out on improving the damping in fibre reinforced plastic (FRP) 
composites as well as the number of approaches which can be taken to improve 
the damping properties of polymeric composites have been summarised in [1]. 
The aim of the research described here was to study lightweight composite 
materials with a highly damped, high temperature polymer matrix material, by 


981 


investigating its mechanical and acoustic fatigue properties, the latter investigation 
being carried out using thin, multilayered plates. 

2-ExperimentaI work 

For this type of study, two adequate prepregs were highlighted after investigation: 
SE300 and PMR15. The SE300 material was carbon fibre reinforced prepreg of 
(0°/90°) woven form, 0.25 mm thick and had 60% fibre volume fraction, with no 
suitable data available on the material properties. Dynamic mechanical thermal 
analyser (DMTA) analyses carried out on four specimens ( 20 nun long and 12 
mm wide) with different lay-ups i.e. (0°/90°)4, (+45°/-45°)4, (0°/90°, 45°/45°)s 
and (45®/45°;0®/90°)s, allowed to get provided the material properties. Results 
from DMTA analyses are shown in Table l.a where the loss factor and the 
Young’s modulus values at 40^C and at the glass transition temperature Tg are 
presented. The loss factor values varied from 0.0097 to 0.085 for a range of 

temperature from 40®C to 300®C. 

The PMR15 prepreg was also of (0^/90^) woven carbon form and had 58% 
volume fraction. Six DMTA specimens having the following lay-ups: (0®/90®)4, 
(00/90^)8, ( 4450 /- 450 ) 4 , (+450/-45<^)8, ( 0 ^/ 90 ^-, 450/450)s and 

( 450 / 450 . 00 / 900 )^ were made. Results from DMTA analyses showed that the loss 
factor values varied from 0.0129 to 0.0857 for a range of temperature from 40®C 
to 400®C, with a value of loss factor of 0.1293 at 375®C, the maximum 
temperature for normal use being 352^C. The loss factor and Young’s modulus 
values at 40®C and Tg are given in Table l.b. 

Mechanical behaviour of the selected materials 

The fatigue characteristics of these new materials were investigated and results 
were compared with those of well established structural materials. Mechanical 
fatigue tests of SE300 and PMR15 samples using "sinusoidal" loading at a chosen 
maximum strain level, i.e. 8000 pS were carried out and performances compared 
to that of an XAS/914 sample. A mechanical (flexural) fatigue rig was used for 
this purpose to test specimens in a cantilevered configuration. Details of the rig are 
available in [2]. The particular clamp used was designed by Drew [3] to induce 
damage in the centre of the specimen instead of having edge damage, i.e. peeling 
while flexural tests are carried out. In order to investigate the performance of these 
new materials at high temperature, fatigue tests were also carried out on samples, 

at 210^C . This was achieved by using a heating system which consisted of two air 


982 



blowers (electronically regulated hot-air guns) positioned at 40 mm above and 
below the specimens, which allowed specimens to be tested at a uniform temperature 

of 210±5°C. The aim of the mechanical fatigue test was to determine the number of 
loading cycles needed for damage to occur and its subsequent growth rate in 
cyclically loaded composite specimens of SE300 and PMR15 matrix materials . The 
samples were 140 mm long, 70 mm wide and 2 mm thick. Fatigue tests of SE300 (S3 
and S4) and PMR15 (PI and P2) specimens, at room temperature and at 210OC 
respectively, at a level of 8000 p.S, located by the peak of the half-sine clamp, have 
been carried out. Ultrasonic scans of specimens S3 and S4 before any loading cycles 
and after 100, 500, 1000, 2000, 5000, 10000, 20000 and 50000 loading cycles are 
shown respectively in Figures l.a-h and Figures 2.a-h. A small delamination, 
indicated by lighter areas in the scans, starts to show in both specimens S3 and S4 
after applying 500 loading cycles and increases substantially after 5000 loading 
cycles. After 5000 loading cycles, the damage area increased more for specimen S4 
than specimen S3, which shows that the latter is slightly more fatigue resistant. In 
other words, when increasing the temperature from 250C to 210<^C, the resistance to 
fatigue slightly decreases. Figures 3.a-h and Figures 4.a-h show the ultrasonic scans 
for PMR15 specimens PI and P2 before and after several loading cycles. For both 
specimens PI and P2, damage starts after 500 loading cycles and increases 
substantially after 20000 loading cycles. At this stage, delamination areas are similar 
for both specimens PI and P2 and just a little more pronounced in specimen P2, 
which shows that the latter is slightly less fatigue resistant. Hence, an increase in 
temperature leads to a decrease in the fatigue resistance properties of the specimens. 
From Figures 1, 2, 3 and 4, one can conclude that the PMR15 specimens tested were 
slightly more fatigue resistant than the SE300 specimens. In fact, damage in 
specimens was generally more defined, clearer and spread more rapidly in the SE300 
samples than was the case for the PMR15 samples. Figure 5 shows damage 
propagation occuring in an XAS/914 sample (XI), with (0o/±45®/0O)s stacking 
sequence, tested at 8000 |iS level and at room temperature, from [3]. Note that 
substantial damage existed after 1000 loading cycles in this specimen, which shows 
that both SE300 and PMR15 are more fatigue resistant than XAS/914 at room 
temperature. 

Acoustic fatigue behaviour of panels of the selected materials 
Investigations were carried out by installing the CFRP plates in an acoustic 
progressive wave tunnel, (APWT) in order to determine the response of CFRP plates 
under broadband acoustic excitation simulating jet noise. The plate was fully 
clamped around its boundaries on to a vertical steel frame fixed to one side of the 
APWT, so that it formed one of the vertical walls of the test section of the APWT. 


983 


Overall sound pressure levels (OSPL) up to 165 dB of broadband noise in the test 
section of the tunnel was generated by a Wyle Laboratories WAS 3000 siren. A 
heater panel capable of heating and maintaining the temperature of test plates up to 

300*^0 while mounted on the tunnel was designed and built. Temperatures were 
monitored and controlled via thermocouples on the panel. Plates were excited by 
broadband excitation in the frequency range 80-800 Hz. A B&K type 4136 
microphone mounted at the centre of the test section of the tunnel adjacent to the 
mid-point of the plate were used for sound pressure measurements. Eight strain 
gauges, four on each side of the plate were attached in order to monitor the strain 
distribution in the panel while the rig was running; more details of the experiment 
may be found in [1]. Acoustic tests were run, at various temperatures and OSPL. 
Since plates were excited in the frequency range 80-800 Hz, spectral analyses 
would not include the first natural frequency. The natural fundamental frequency of 
an SE300 clamped panel was found to be 49.02 Hz analytically. The second -and 
third resonance frequencies were 149.5 and 198.5 Hz, as calculated from strain 
spectral densities, from tests carried out with an OSPL of 156 dB and a temperature 
of 1620C. At 162 dB, results showed that the second resonance frequency was 113 
Hz at 150OC and 106.5 Hz at 195^0, which shows that when the temperature and 
the OSPL increased, the resonance frequencies of the plates decreased. Also, it was 
found that the damping increased at elevated temperatures. The overall modal 
viscous damping ratios, for the second mode, were calculated from strain spectral 
densities, for an SE300 panel driven at an OSPL of 162 dB and at 1950C, and was 
found to be 8.91%; this value is similar to that calculated from analytical 
simulations, for the first mode, which is 8,50% at 242^0, (see the analytical section 
below). A typical strain spectral density obtained from recorded results is presented 
in Figure 6 for an SE300 specimen, at an OSPL = 156 dB and at 1620C, from a 
strain gauge in the centre of the specimen. Maximum RMS strain values recorded 
from experimental tests, at a strain gauge in the centre of the specimen were, at an 
OPSL =156 dB, 1300 \iS, 1800 \iS and 2800 pS at room temperature and at 90^0 
and 1620C respectively. These results clearly indicate a trend for significant 
increase in dynamic response with increasing temperature. 

Experiments were also carried out on a PMR15 panel at various OSPL and 
temperatures. Results from tests run at 159 dB and at room temperature show that 
the second and third resonance frequencies were indicated as 112 and 182.5 Hz. 
When the OSPL increased to 162 dB, the second and third resonance frequencies 
decreased to 110.5 Hz and 176.5 Hz. At the same OSPL (162dB) and when 
temperature increased to 2810C, the third resonance frequency became 139 Hz. This 


984 


shows that, for PMR15 plates, when the temperature and the OSPL increased, the 
resonance frequencies of the plates decreased. It was also clear that modal damping 
increased with increasing temperature. In fact, the overall viscous damping reached 
20%, for the second mode, at an OSPL=162 dB and at 2810C. It must be stated here, 
however, that apparent damping trends could include nonlinear effects which 
influence bandwidths of resonances. Maximum RMS strain values recorded at room 
temperature, by a strain gauge in the centre of the plate were found to be: 2700 pS at 
153 dB, 2800 pS at 157.9 dB and 2900 pS at 159 dB. 

It is clear from these values that increasing the OSPL obviously leads to an increase 
of the strain in the plate. Similar results were observed when the temperature was 
increased. In fact, at an OSPL of 162 dB, the maximum strain values recorded by a 
strain gauge in the centre of the specimen were 3000 pS, 3400 pS and 5000 pS at 
105^0, 1650C and 281^0 respectively, which clearly indicates the effects of 
temperature. It was observed that both the PMR15 and SE300 panels behaved in a 
non linear manner. 

Attempts to acoustically fatigue a PMR15 panel were made at 162 dB. No signs of 
fatigue damage were shown in an ultrasonic scan of the panel after 1389 minutes of 
running time. 


3-Analytical work 

In order to examine various methods for including damping in a structure, parametric 
studies were carried out using the finite element FE method. ANSYS software has 
been used. A three-dimensional, 3D layered element, SOLID46 was used to build 
theoretical models. The element is defined by eight nodal points, average layer 
thickness, layer material direction angles and orthotropic material properties, [4]. 
Meshes were built in order to carry out modal and harmonic analyses of multilayered 
composite plates (410 mm, 280 mm, 2 mm). The plates were fully clamped along all 
edges, in order to simulate the panels tested in the APWT. Natural frequencies were 
first determined from free vibration analyses and compared to resonance frequency 
values derived from experimental data. Then, the plate was driven by harmonic 
loading at one point of application. The forcing frequency varied from 0 to 400 Hz. 
The amplitude of the load was 50 N. Results for displacements and response phase 
angles relative to the force for a chosen position on the plate as a function of 
frequency were obtained. The approach was then to carry out parameter studies in 
order to examine various methods for including damping in the structure, i.e. to use 
highly damped matrix material throughout the whole structure or possible 


985 



incorporation in a few layers. Structural damping was included, allowing models to 
run with different damping values in each ply of the panel. Structural damping is 
inherent in the structure and depends on the natural frequency; details on structural 
damping modeling may be found in [1]. Analyses were performed considering 
structural damping for the first mode. The structural damping was then varied for 
plies with the same orientation for a viscous damping ratio ^ = 0.01,0.02, 0.05, 0.10 
and 0.20. 

Simulations with SE300 

Models were built up with the following stacking sequence ((45°/45°),(0°/90°))s, 
lay-up used for the experimental plates. Table 2 gives the first three modal 
frequencies of the panel obtained from free vibration analyses results. Harmonic 
simulations were carried out and the overall damping value was calculated for each 
case with results given in Table 3. As can be seen, if high overall damping is needed 
for a structure composed of the SE300 material, increasing the damping value of the 
(45°/45®) orientation plies most significantly increases the overall damping value of 
the panel. In fact, putting a damping value of 20% in the (45°/45°) orientation plies 
leads to an overall viscous damping value of 14.52%, which is better than including 
a 10% damping value in all of the plies of the structure. 

Harmonic analyses of fully clamped plates were also performed with the values of 
material properties taken at several temperatures. Simulations were carried out with 
material properties at 2420C and 300OC. Free vibration analyses permitted 
calculation of the modal frequencies of the panels at the temperatures mentioned 
above. Table 2 also lists the first three modal frequencies from analyses with 
material properties at 242^0 and 300^0. The overall viscous damping values, 
obtained from FE simulations, are given for each ternperamre in Table 4. Again, the 
damping value has been varied through the layers and the overall damping value was 
calculated in order to see which of the plies contributes the most to heavily damp the 
plate. It was found that putting a damping value of 20% in the (45°/45°) orientation 
plies, the first mode viscous damping ratios were 14.62% and 14.55% at 242^0 and 
300OC respectively. This shows that this material is more highly damped at high 
temperature and presents better damping properties of the two materials at 242^0. 

Simulations with PMR15 

Free vibration analyses of models built up with the following stacking sequence 
((45°/45°),(0°/90®))s were carried out and the first three modal frequencies of the 
panel are shown in Table 5. Harmonic analyses were run and the overall damping 


986 



value was calculated for each simulation with results given in Table 6. If high 
overall damping is needed for a structure composed of the PMR15 material, 
increasing the damping value of the (45°/45®) orientation plies most significantly 
increases the overall damping value of the panel. In fact, putting a damping value 
of 20% in the (45°/45°) orientation plies leads to an overall viscous damping value 
of 14.39%, while if (0°/90®) orientation plies have a 20% damping value, the 
overall damping is 7.42%. 

Harmonic analyses of fully clamped plates were also carried out with the values of 
material properties taken at several temperatures. Simulations were carried out 
using material properties at 3750C and 400oC. Free vibration analyses permitted 
calculation of the modal frequencies of the panels at the temperatures mentioned 
above. Table 5 lists the first three modal frequencies from analyses with material 
properties at 3750C and 400^0. The overall viscous damping values, obtained 
from FE simulations, are given for each temperature in Table 7. Again, the 
damping value has been varied through the layers and the overall damping value 
was calculated in order to see which of the plies contributes the most to heavily 

damp the plate. It was found that putting a damping value of 20% in the (45‘^/45°) 
orientation plies, the first mode viscous damping ratios were 18.39% and 16.94% 
at 3750 c and 400OC respectively. This shows that this material is more highly 
damped at high temperature and presents better damping properties of the two 

materials at 3750C. 


4~Conclusions 

Two matrix materials, SE300 and PMR15, with potential for use in aircraft 

structures in a severe environment, i.e. temperatures up to SOO^C were selected for 
this study. Material properties were determined using DMTA techniques and 
results show that these materials have high damping abilities at high temperature. 
Dynamic loading tests, performed in flexure at room and high temperature showed 
that the carbon fibre reinforced PMR15 material is more fatigue resistant than 
SE300 and XAS/914 based composites. Acoustic tests using a progressive wave 
tunnel, up to a random acoustic OSPL of 162 dB, at room temperature and 
elevated temperamres up to 2810C were also performed. When increasing the 
excitation level and the temperature higher strain values in the centre of the panels 
were recorded. Free vibration and harmonic FE analyses permitted determination 
of the natural frequencies and the overall viscous damping values. Resonance 
frequencies determined from results obtained from acoustic tests were similar to 


987 


natural frequencies obtained from FE simulations. Overall viscous damping values 
obtained from experimental results agreed well with those obtained from the FE 
analyses for SE300 panels. Results obtained for PMR15 panels, from tests, were 
higher than those calculated analytically. Both tests and simulations showed that 
SE300 and PMR15 present higher damping capabilies at high temperatures. 
Conclusions, via parameter studies including material damping, have been drawn 
concerning optimal incorporation of a highly damped matrix material into a high 
performance structure. 


5-AcknowIegments 

The authors wish to thank the Minister of Defence for sponsorship of the programme 
of research under which the work was carried out. Thanks are also due to Dr M. Nash 
of the DRA, Famborough for many helpful discussions throughout the project. _ 

6-References 

1- Benchekchou, B. and White, R.G., Acoustic fatigue and damping technology in 
FRP composites, submitted to Composite Structures. 

2- Benchekchou, B. and White, R.G., Stresses around fasteners in composite 
structures in flexure and effects on fatigue damage initiation: I-Cheese-head bolts. 
Composite structures, 33(2), pp. 95-108, November 1995. 

3- Drew, R.C. and White, R.G., An experimental investigation into damage 
propagation and its effects upon dynamic properties in CFRP composite material . 
Proceedings of the Fourth International Conference on Composite Structures, Paisley 
College of Technology, July 1987. 

4- ANSYS theoretical manual, Swanson Analysis Systems Inc, December 1992. 


988 


Table La: Loss factor and Young’s modulus values at 40^C and at Tg for 
SE300 samples analysed by the DMTA. 



stacking sequences 

(4.45/-45)4 

(-457+45)4 

(45/45;0/90)s 

(0/90)4 

Tg(°C) 

242 

242 

240.3 

238.71 

T| atTg 

0.085 

0.085 

0.081 

0.061 

Ti at 40°C 

0.012 

0.014 

0.010 

0.0097 

Log E’ at40°C 

9.870 

9.840 

9.970 

10:097 


Table Lb: Loss factor and Young’s modulus values at 40^C and at Tg for 
PMR15 samples analysed by the DMTA. 



Stacking sequences 

(+45/--45)4 

(0/90;45/45)s 

(45/45;0/90)s 

(0/90)4 

Tg (OC) 

372 

375 

375 

375 

Tl atTg 

0.117 

0.124 

0.121 

0.129 

T| at 40®C 

0.0110 

0.0138 

0.0086 

0.0117 

Log E’ at 40OC 

9.583 

9.944 

9.972 

9.875 


989 































Table 2: The first three modal frequencies for SE300 panel; analyses carried out 
with material propert ies at room temperature, at 242*^0 and at 300°C. 


Room temperature 

242^0 

300OC 

49.02Hz 

44.72Hz 

43.61Hz 

155.80Hz 

143.27Hz 

140.17Hz 

212.04Hz 

194.19Hz 

189.68Hz 


Table 3: Overall viscous damping values of SE300 panel. Values are calculated from 
results obtained from harmonic analyses; the material damping being considered for 
the first mode. 


Simulation with 
damping of 

(45°/45°)orientation 

plies 

(0®/90°)orientation 

plies 

5% 

3.81% 

2.51% 

10% 

7.06% 

3.92% 

20% 

14.52% 

11.57% 


Table 4: Overall viscous damping values of SE300 panel, for the first mode. 


Temperature 

(°C) 

Overall viscous damping 

25 

1.20% 

242 

8.50% 

300 

5.45% 


990 



















Table 5: The first three modal frequencies for PMR15 panel; FE analyses carried out 
with material properties at room temperature, 375°C and 400°C. 


Room temperature 

3750c 

400OC 

43.43Hz 

32.68Hz 

25.54Hz 

130.35Hz 

99.65Hz 

78.83Hz 

183.96Hz 

139.04Hz 

109.11Hz 


Table 6; Overall viscous damping values of PMR15 panel. Values are calculated 
from results obtained from harmonic analyses; the material damping being 
considered for the first mode.. 


Simulation with 
damping of 

(45°/450)orientation 

plies 

(0o/90O)orientation 

plies 

5% 

3.76% 

2.44% 

10% 

7.26% 

4.10% 

20% 

14.39% 

7.42% 


Table 7: Overall viscous damping values of PMR15 panel, for the first mode.. 


Temperature 

Overall viscous damping 

(OQ 


25 

1.33% 

275 

13.24% 

400 

8.6% 


991 




























a: before any loading cycles b: after 100 loading cycles c: after 500 loading cycles 



d: after 1000 loading cycles e: after 2000 loading cycles f: after 5000 loading cycles 



g: after 10000 loading cycles h: after 20000 loading cycles 


Figure 1. Ultrasonic scans of specimen S3 after applying different numbers of loading cycles. 
(SE 300 material, ambient temperature) 



a: before any loading cycles b: after 100 loading cycles c: after 500 loading cycles 



d: after 1000 loading cycles e: after 2000 loading cycles f: after 5000 loading cycles 



Fisure 2. Ultrasonic scans of specimen S4 after applying different numbers of loading cycles. 

(SE300 material, 210OC) 


992 




















a: before any loading cycles b: after 100 loading cycles c: after 500 loading cycles 





d: after 2000 loading cycles e: after 20000 loading cycles f: after 50000 loading cycles 




g: after 100000 loading cycles 


Figure 3. Ultrasonic scans of specimen PI after applying different numbers of loading cycles 
(PMR15 material, ambient temperature, 8000 ]LlS) 




a: before any loading cycles b: after 500 loading cycles c: after 1000 loading cycles 



g: after 10000 loading cycles 


h: after 20000 loading cycles 


Figure 4. Ultrasonic scans of specimen P2 after applying different numbers of loading cycles. 

(PMR15 material, 2100C, 8000M.S) 


993 



































a: before any b: after 100 c: after 500 d: after 1000 

loading cycles loading cycles loading cycles loading cycles 



e: after 2000 f: after 5000 g: after 10000 h: after 20000 

loading cycles loading cycles loading cycles loading cycles 

Figure 5. Ultrasonic scans of an XAS/914 specimen fatigued at a level of 8000 llS 
showing the damage propagation; the lay-up is (0/±45/90)s, [3], 



0 Lin Hz RCLD 1.6k 


Figure 6: SE300 specimen SI strain spectral density, recorded from strain gauge 
ST2, OSPL =156 dB, temperature = 162^C. 


994 






























THE BEHAVIOUR OF LIGHT WEIGHT HONEYCOMB SANDWICH 
PANELS UNDER ACOUSTIC LOADING 


David Millar 
Senior Stress Engineer 
Short Bros. PLC 
Airport Road 
Belfast 

Northern Ireland 


SUMMARY 

This paper discusses the results of a progressive wave tube test on a carbon composite 
honeycomb sandwich panel. A comparison was made with the test panel failure and the 
failure of panels of similar construction used in the intake ducts of jet engine nacelles. 
The measured panel response is compared with traditional analytical methods and finite 
element techniques. 


Nomenclature 

= Overall rms stress (psi) or strain (jxs). 

7t =3.14159 

= Fundamental frequency (Hz). 

5 = Critical damping ratio (*0.017). 

Lps(fn) = Spectrum level of acoustic pressure (- expressed as a fluctuating rms pressure in psi in a 
1 Hz band). 

jr = Joint acceptance function (non dimensional). 

= Characteristic modal pressure (psi) 
ph = Mass per unit area (Ib/in^) 

Sic “ Modal stress (psi) or modal strain (|j£). 

Wjc = Modal displacement (in), 

a = Panel length (in), 

b = Panel width (in). 

x,y,z = Co-ordinate axes. 


1.0 Introduction 

Honeycomb sandwich panels have been used for some time in the aircraft industry as 
structural members which offer a high bending stiffness relative to their weight. In 
particular, they have proved very attractive in the construction of jet engine nacelle 
intake ducts where, in addition to their load carrying ability, they have been used for 
noise attenuation. 


995 


2.0 Acoustic Fatigue 

The intake duct of a jet engine nacelle can experience a severe acoustic environment 
and as such the integrity of the nacelle must be assessed with regard to acoustic fatigue 
[1], Acoustic fatigue characterises the behaviour of structures subject to acoustic 
loading, in which the fluctuating sound pressure levels can lead to a fatigue failure of 
the structure. The traditional approach to acoustic fatigue analysis has assumed 
fundamental mode response and given that aircraft panels will in general, have 
fundamental frequencies of the order of several hundred hertz, it is clear that the 
potential to accumulate several thousand fatigue cycles per flight can exist. 

Techniques for analysing the response of structures to acoustic loads were developed 
originally by Miles [2] and Powell [3], Other significant contributions are listed in 
References 4-7. Design guides such as AGARD [8] and the Engineering Sciences 
Data Unit (ESDU) series of data sheets on vibration and acoustic fatigue [9], have 
proved useful in the early stages of design. 

Note - further details on the general subject of acoustic fatigue can be found in Ref 
10, while a more detailed review of the subject up to more recent times is presented in 
Ref 11. 


3.0 In Service Failures 

In recent years a number of failures have been experienced involving intake barrel 
honeycomb sandwich panels. Failures have been experienced with panels which had 
both aluminium facing and backing skins and carbon composite panels. The metal 
intake liner was observed to have skin cracking and also core failure, while the 
composite panel was only observed to have core failure. 

With regard to the metal panels, flight testing was carried out and the predominant 
response frequency was observed to be at the fan blade passing frequency - much 
higher than the fundamental frequency of the intake barrel; this went some way to 
explaining why the traditional approach in estimating the response did not indicate a 
cause for concern. The response of the panel was also very narrow band - almost a 
pure sinusoid (again differing from the traditional approach of broad band/random load 
and response), and the subsequent analysis of the results was based on a mechanical 
fatigue approach [12]. Subsequent fleet inspections revealed that core failure was 
observed prior to skin failure and it was assumed that the skin failure was in fact 
caused by a breakdown in the sandwich panel construction. The core was replaced 
with a higher density variety, with higher shear strength and moduli. This modification 
has been in service for several years with no reported failures. The modification 
represented only a moderate weight increase of the panel, without recourse to 
changing skin thickness, which would have proved very expensive and resulted in a 
substantial weight penalty. 

As mentioned above, another intake duct, of carbon composite construction, also 
began to suffer from core failure. The panels of this duct had a carbon backing skin 
while the facing skin had a wire mesh bonded to an open weave carbon sheet. The only 
similarity was the use of the same density of honeycomb core (although of different 
cell size and depth). For other reasons this core had been replaced by a heavier variety, 
prior to the discovery of the core failures and the impact of the failures was minimised. 


996 


Limited data is available on similar failures and only 2 other cases, regarding nacelle 
intake barrels, appear to have been documented [13 & 14], however neither case 
involved sandwich panels. 

A number of theories had been put forward as to the cause of the failures. These 
included neighbouring cells resonating out of phase, cell walls resonating or possibly 
the panel vibrating as a 2 degree of freedom system (the facing and backing skins 
acting as the masses, with the core as the spring) - this phenomenon had originally 
been investigated by Mead [15]. 


4.0 Physical Testing 

A number of tests were carried out with "beam" type high cycle fatigue specimens and 
also small segments of intake barrel. None of these tests were able to reproduce the 
failures observed in service (Figure 1.0) which further served to reinforce the belief 
that the failures were attributed to an acoustic mechanism as opposed to a mechanical 
vibration mechanism, however in an attempt to cover all aspects it was decided to 
carry out a progressive wave tube (PWT) test on an abbreviated panel. 

For simplicity it was decided to test a flat sandwich panel of overall dimensions 
36"x21" (Figure 2.0). The panel was instrumented with 12 strain gauges and 2 
accelerometers. Two pressure transducers were also mounted in the fixture 
surrounding the specimen. 

Testing was carried out by the Consultancy Service at the Institute of Sound and 
Vibration Research (IS VR) at the University of Southampton. 


4.1 PWT Results 

The panel was first subject to a sine sweep from 50 to 1000 Hz in order to identify its 
resonant frequencies. The response of a strain gauge at the centre of the panel has been 
included in Figure 3.0. On completion of the sine sweeps, the linearity tests were 
carried out.. As only 8 channels could be accommodated at one time, it had been 
decided to arrange the parameters into 5 groups, with each group containing 4 strain 
gauges, 2 accelerometers, 1 pressure transducer on the fixture and 1 pressure 
transducer in the PWT (this was required by the facility for the feedback loop). 


The initial tests were carried out with a power spectral density of the applied loading 
constant over the 100 Hz to 500 Hz range, however when using this bandwidth only 
155 dB overall, could be achieved. In an attempt to increase the strain levels it was 
decided to reduce the bandwidth to 200 Hz. The bandwidth (BW) was subsequently 
reduced to 100 Hz and finally 1/3 octave centred on the predominant response 
frequency of the panel. When failure occurred a dramatic change in response was 
observed. The failure mechanism was that of core failure as shown in Figure 4.0. There 
was no indication of facing or backing skin distress. 


997 



5.0 Comparison With Theoretical Predictions 
5.1 Fundamental Frequency 

From the strain gauge readings the panel was seen to be vibrating with simply 
supported edge conditions. Soovere [7] suggests that "effective" dimensions 
(essentially from the start of the pan down) be used to determine the fundamental 
frequency which is given by; 


x\n 



This equation is applicable to simply supported panels with isotropic facing and 
backing skins, thus for the purpose of applying the above equation, the actual section 
was approximated to a symmetric (isotropic) section. The predicted fundamental 
frequency is given below. It was observed however, that if the panel dimensions are 
taken relative to mid way between the staggered pitch of the fasteners a significant 
improvement was achieved (see "Soovere (2)" in table 1). Alternative frequency 
estimations using an FE model and an ESDU data item [16] are summarised in the 
following table; 


Method 

Freq.(Hz) 

% Error 

Measured 

228 

- 

Soovere 

274.3 

+20.3 

Soovere 

(2) 

213.3 

-6.4 

FE 

239.04 

+4.8 

ESDU 

193 

-15.3 


Table 1 - Comparison of calculated frequencies for simply supported sandwich 

panel. 

Note; the percentage error is based on the actual measured response frequency of the 
panel in the PWT. 

Given that the excitation bandwidth extended (at least initially) up to 500 Hz, modes 
up to 500 Hz were obtained from the FE model. In actual fact 2 FE models were used, 
the first was a basic model with 380 elements, however a more detailed model, shown 
in Figure 5.0, (with essentially each element split into 4) was used for the results 
presented in this paper. The predicted modes from the FE model were as follows; 


Mode 

No. 

Frequency 

(Hz) 

Mode No. in x 
direction (m) 

Mode No. in y 
direction (n) 

Figure 

No. 

1 

239.04 

1 

1 

6 

2 

334.0 

2 

1 

7 

3 

430.02 

1 

2 

8 

4 



1 

9 


Table 2 - Finite Element Model Predicted Frequencies. 


998 










5.2 rms Strain 

The predicted strains were calculated using Blevins' normal mode method (NMM) [5], 
with a joint acceptance of unity for the fundamental mode of vibration, using the 
following expression; 


s^= Lp.(fJ . ^ (2) 

In an attempt to improve the estimated response, the rms strain was calculated for 
each mode within the bandwidth of excitation. The Joint accetptance for each mode 
was calculated using equation 3 and the calculated strains for each mode were then 
factored by the relevant joint acceltance term. The overall strain was then calculated 
for all the relevant modes. A comparison with ESDI! [16] has also been included, 
however the ESDU method does not provide an indication of shear stress in the core. 

Soovere presents a simple expression for the joint acceptance function for a simply 
supported panel excited by an (acoustic) progressive wave, for the case where n is 
odd; 

•2 _ ^ (l-Cos(m7c)Cos(c0ra/c) , . 

(1 - (cD^a/mrcc)^) 

Note, when n is even the joint acceptance is zero. 

Given that the bandwidth varied for the applied loading, the overall SPLs were 
expressed as spectrum levels for the purpose of comparison in the linearity results, the 
results (both measured and predicted) have been summarised in table 3, (SGI results 
have been plotted in Figure 10.0). The results from the ESDU data item [16] have 
been included in table 4 for comparison. 

Note - due to recorder channel limitations SGI & SG2 were not connected at the time 
of failure and no results were available at the highest sound pressure levels. 


OASPL 

(dB) 

Spectrum 
Level SPL 
(dB) 

Measured 
Strains(u£) 
SGI SG2 

Calculated 
(ps) j=l 

SGI SG2 

Calculated 
(Multi Mode) 
SGI (us) SG2 

130 

107 

7 

7 

8.7 

9.2 

2.5 

2.6 

140 

117 

20 

19 

27.5 

29.1 

8.0 

8.1 

150 

127 

55 

60 

87.0 

92.0 

25.4 

25.8 

155 

132 

100 

100 

154.7 

163.7 

45.1 

45.8 

157 

134 

130 

130 

194.8 

206.0 

56.8 

57.6 

163 

140 

- 

- 

388.7 

411.1 

202.5 

162.7 

164 

141 

- 

- 

436.1 

461.2 

312.0 

250.7 


Table 3 - Comparison of Measured & Predicted rms Strains 
for the Panel Centre, Facing & Backing Skin Gauges. 


999 



OASPL 

(dB) 

Spectrum 
Level SPL 
(dB) 

Measured 
Strains (pe) 
SGI SG2 

ESDI! Strains 

SGI SG2 

130 

107 

7 

7 

10.1 

29 

140 

117 

20 

19 



150 

127 

55 

60 

101 

290 

155 

132 

100 


179.5 

515.7 

157 

134 

130 

■E9 

253.6 

728.5 

163 

140 

- 

- 

637 

1830 

164 

141 

- 

- 

1010 

2900 


Table 4 - Comparison of Measured & ESDI! Predictions of the 
rms Strains for the Panel Centre, Facing & Backing Skin Gauges. 

There is a considerable difference in the calculated response from using a joint 
acceptance of unity for the fundamental mode and that when estimating the joint 
acceptance for each mode and calculating the overall response for several modes, 
however it was observed that if the average value from both methods is used the 
response compares favourably with that measured (-at least for the cases under 
consideration). The average value has been included on the linearity plot for SGI, 
shown in Figure 10.0).In general, the level of agreement between theory and practice 
was considered adequate and it was decided to apply the theory to estimating the shear 
stresses in the core (Table 5); 


OASPL 

(dB) 

Spectrum 
Level SPL 
(dB) 

Core Shear 
Stress (J=l) 
(rms psi) 

Core Shear 
Stress 

(Multi Mode) 
(rms psi) 

Average 
Core Shear 
Stress 
(rms psi) 

Peak Core 
Shear 
Stress 

_(E£!i_ 

130 

107 

0.3 

0.09 

0.19 

0.58 

140 

117 

0.94 

0.28 


1.84 

150 

127 

2.99 

0.88 

1.93 

5.8 

155 


5.31 

1.57 

3.44 

10.32 

157 

134 

6.69 

1.97 

4.33 

13.0 

163 

140 

13.35 

5.57 

9.46 

28.38 

164 

141 

14.98 

8.59 

11.78 

35.34 


Table 5 - Predicted Core Shear Stress. 

6.0 Discussion & Recommendations 

The ESDI! method proved very conservative and will thus give a degree of 
confidence when used in the early stages of the design process. Blevins Normal Mode 
Method was observed to give reasonable accuracy in predicting the highest strains in 
the panel and would merit use when designs have been fixed to some degree; at which 
stage FE models become available. 

For panels whose predominant response is in the fundamental mode it is accepted that 
the contribution from shear to overall deformation is very small. The main concern 
when designing a honeycomb sandwich panel which is subject to "severe" acoustic 


1000 






























































loads has tended to focus on skin strains and to some degree the properties of the core 
material have been ignored. The fact that low skin strains are observed has the effect of 
giving an impression that there is no cause for concern, however when the properties 
of the core material are low or unknown, some caution is required. There is 
unfortunately no available S-N data for the type of honeycomb used in the construction 
of the panel, however the allowable ultimate strength for the core material is of the 
order of 26 psi, so clearly the 163 dB level was sufficient to cause a static failure while 
the lower SPLs can be assumed to the have contributed to initiating fatigue damage. 

On cutting up the test panel, a large disbond was observed however it did not extend 
to the panel edge where cracking had occurred (the mid point of the long edge being 
the location of maximum shear for a simply supported panel) and it was the opinion of 
the materials department that the failure had not initiated in the disbond. 

The SPLs used in the test were not excessively high and were comparable to service 
environments (an example of which is given in Table 6). It should be noted that while 
the levels in Table 5 are 1/3 octave bandwidths, the actual spectrum is not generally 
flat within each band for engine intakes, but is rather made up of tones (Figure 11). 
These tones or spectrum levels can thus essentially be the band level and thus some 
caution should be exercised when converting intake band levels to spectrum levels 
using the traditional approach [17]. 


1/3 Octave Centre 

Sound Pressure 

Frequency (Hz) 

Level (dB) 

100 

141 

125 

133 

160 

140 

200 

142 

250 

140 

315 

139 


Table 6 - Typical Acoustic Service Environment. 

Note; Overall levels may reach 160 - 170 dB, however they tend to be influenced by 
SPLs at blade passing frequencies, which are much higher than panel fundamental 
frequencies. 


7.0 Conclusion 

It has been shown that although moderate levels of acoustic excitation produce quite 
low overall rms strains in the skins of honeycomb sandwich panels, it is still possible, 
when using very light weight cores, to generate core shear stresses of a similar order of 
magnitude to the allowable ultimate strength of the material. 


Acknowledgements 

The author acknowledges the support of Short Bros. PLC in the course of preparing 
this paper and also the assistance of Mr Neil McWilliam with regard to the FE 
modelling. 


1001 




References 


I. 0 Air worthiness Requirements (JAR/FAR) Section 25.571 .d. 

2.0 Miles, J.W., "On Stmctural Fatigue Under Random Loading," Journal of the 
Aeronautical Sciences, (1954),Vol.21, p753 - 762. 

3.0 Powell, A., "On the Fatigue Failure of Structures due to Vibrations Excited by 
Random Pressure Fields,” Journal of the Acoustical Society of America, (1958), 
Vol.30, No.l2,pll30- 1135. 

4.0 Clarkson, B.L., "Stresses in Skin Panels Subjected to Random Acoustic 
Loading," Journal of the Royal Aeronautical Society, (1968), Vol.72, 
plOOO- 1010. 

5.0 Blevins, R.D., "An Approximate Method for Sonic Fatigue Analysis of 
Plates & Shells," Journal of Sound & Vibration, (1989), Vol.129, No.l, 
p51-71. 

6.0 Holehouse, I., "Sonic Fatigue Design Techniques for Advanced Composite 
Aircraft Structures," AFWAL TR 80-3019,(1980). 

7.0 Soovere, J., "Random Vibration Analysis of Stiffened Honeycomb Panels with 
Beveled Edges," Journal of Aircraft, (1986), Vol.23, No.6, p537-544. 

8.0 Acoustic Fatigue Design Data (Part 1), AGARD-AG-162-72, (1972). 

9.0 ESDU International, London, Series on Vibration & Acoustic Fatigue. 

10.0 Richards, E.J., Mead, D.J.,"Noise and Acoustic Fatigue in Aeronautics," John 
Wiley & Sons, New York, (1968). 

II. 0 Clarkson, B.L., "A Review of Sonic Fatigue Technology," NASA CR 4587, 

(1995). 

12.0 Millar, D., "Analysis of a Honeycomb Sandwich Panel Failure," M.Sc. Thesis, 
University of Sheffield, (1995). 

13.0 Holehouse, L, "Sonic Fatigue of Aircraft Structures due to Jet Engine Fan 
Noise," Journal of Sound & Vibration, (1971), Vol. 17, No.3, p287-298. 

14.0 Soovere, J., "Correlation of Sonic Fatigue Failures in Large Fan Engine Ducts 
with Simplified Theory," AGAEUD CPI 13 (Symposium on Acoustic Fatigue), 
(1972), pi 1-1 - 11-13. 

15.0 Mead, D.J., "Bond Stresses in a Randomly Vibrating Sandwich Plate: Single 
Mode Theoiy," Journal of Sound & Vibration, (1964), Vol.l, No.3, 
p258-269. 

16.0 ESDU Data Item 86024 (ESDUpac A8624), "Estimation of RMS Strain in 
Laminated Face Plates of Simply Supported Sandwich Panels Subjected to 
Random Acoustic Loading," Vol. 3 of Vibration & Acoustic Fatigue Series. 

17.0 ESDU Data Item 66016, "Bandwidth Correction," Vol. 1 of Vibration & 
Acoustic Fatigue Series. 


1002 



Backing Skin 



Figure 2.0 - PWT Test Specimen. 


1003 



















Microstrain (dB) 



Figure 3.0 - Response of Strain Gauge SGI During Sine Sweep. 



Figure 4.0 - Section Through Failure Region in PWT Panel. 


1004 



















Figure 5.0 - PWT Panel Finite Element Model. 



Figure 8.0 - FE Mode 1 (m==l, n=2) Figure 9.0 - FE Mode 2 (m=3, n=l) 


1005 








Sound Pressure Level (dB) 


Figure 10.0 - Linearity Plot for SGI (Measured & NMM Prediction). 



Figure 11.0 - Typical Spectral Content of Intake Duct Sound 
Pressures with Equivalent 1/3 Octave Levels Superimposed. 


1006 










Time Domain Dynamic Finite Element Modelling in Acoustic Fatigue Design 


Authors: 

P. D. Green 
Military Aircraft 
British Aerospace 
Warton 

A. Killey 

Sowerby Research Centre 
British Aerospace 
Filton 

Summary 

Advanced Aircraft are expected to fly in increasingly severe and varied acoustic environments. 
Improvements are needed in the methods used to design aircraft against acoustic fatigue. Since fatigue 
life depends strongly on the magnitude of the cyclic stress and the mean stress, it is important to be able 
to the predict the dynamic stress response of an aircraft to random acoustic loading as accurately as 
possible. 

The established method of determining fatigue life relies on linear vibration theory and assumes that the 
acoustic pressure is fully spatially correlated across the whole structure. The technique becomes 
increasingly unsatisfactory when geometric non-linearities start to occur at lugh noise levels and/or 
when the structure is significantly curved. Also the excitation is generally not in phase across the whole 
structure because of complex aerodynamic effects. 

Recent advances in finite element modelling, combined with the general availability of extremely fast 
supercomputers, have made it practical to carry out non-linear random vibration response predictions 
using time stepping finite element (FE) codes. 

Using the time domain Monte Carlo (TDMC) technique it is possible to model multi-modal vibrations 
of stiffened aircraft panels without making the simplifying assumptions concerning the linearity of the 
response and the characteristics of the noise excitation. 

The technique has been developed initially using a simple flat plate model. This paper presents some of 
the results obtained during the course of this work. Also described are the results of a study of the 
“snap-through” behaviour of the flat plate, using time domain finite element analysis. For simplicity, it 
was assumed that the dynamic loading was fully in phase across the plate. 


Introduction 

Aircraft structures basically consist of thin, generally curved, plates attached to a supporting framework. 
During flight these stiffen^ panels are subjected to a combination of static and dynamic aerodynamic 
loads. On some aircraft there may be additional quasi-static thermal loads due to the impingement of jet 
effluxes in some areas. Parts of advanced short take of and landing (ASTOVL) .aircraft may be required 
to withstand noise levels up to 175dB and temperatures up to 200deg C. Under these conditions the 
established methods of dynamic stress analysis for acoustic fatigue design are inappropriate and cannot 
be employed. 

British Aerospace (BAe), Sowerby Research Centre (SRC) and Military Aircraft (MA) have been 
developing a method to predict the stress/strain response of aircraft structures in these extreme loading 
situations. The primary .consideration has been the requirement to create an acoustic fatigue design tool 
for dealing with combined static and dynamic loads, including thermally generated “quasi-static” loads. 


1007 


The resonant response of thin aircraft structures to aeroacoustic loading is generally in a firequency 
range which implies that, if defects form, they will quickly grow. Hence to be conservative, it is 
generally assumed that a component has reached its life when it is possible to find quite small defects 
by non-destructive evaluation techniques. Several different materials and construction methods are used 
in modern aircraft and so there are a number of possible failure criteria. In the case of metals, it is the 
presence of cracks larger than a certain size. For composites it can be the occurrence of either cracking 
or delamination. Degradation due to the presence of microcracks may be monitored by measuring the 
level of stiffness reduction which has taken place. 

This philosophy simplifies the type of stress analysis needed, because it is not necessary to model 
structures with defects present. Materials can be assumed to have simple elastic properties which 
remain unchanged throughout their lives. In consequence, it is necessary to know the fatigue behaviour 
in terms of a direct relationship between number of cycles to failure and the magnitude of the “nominal” 
cyclic stress, or strain, at a reference location. 

If considered important and capable of satisfactory treatment, the relationship can be modified to take 
into account material property changes due to the development of very small defects at points of stress 
concentration. For example, metal plasticity in the region of a small crack, could be included in an 
analysis of the stress distribution around a fastener hole. It is well known that plasticity reduces the 
peaks of stress which are predicted at defects by analysis which assumes perfectly elastic material 
behaviour. 

The technique developed at BAe for modelling high acoustic loads combined with possible thermal 
buckling uses the time domain Monte Carlo (TDMC) technique together with finite element analysis by 
proprietary FE codes. Response characteristics are predicted directly in the time domain using 
simulated random acoustic loadings. These may then be used in fatigue life estimations which employ 
cycle counting methods such as Rainflow counting. It is now practical to predict the vibrational 
response of stiffened aircraft panels without the necessity to assume a linear response, and without 
simplifying the spatial and temporal representation of the noise excitation. 

Since the technique uses proprietary finite element codes, quite large and complex models of aircraft 
structure can be analysed in a single run. Standard pre- and post-processor techniques are available to 
speed up generation of the finite element mesh and to display the stress/strain results. 

The initial development work was carried out by modelling the random vibration of a flat plate. For 
fully in-phase random loading at low noise levels the predicted response is predominantly single mode 
and at fhe frequency calculated by linear theory. However, as the decibel level is increased, the 
frequency of the fundamental rises due to geometrically non-linear stiffening. At veiy high dB levels the 
predicted response becomes multi-modal; the resonance peaks move to higher and higher frequencies 
and broaden. 

The effect of static loading on the response has been studied as part of these investigations to assist in 
the validation of the methodology being developed. 


Thermal Effects 

In some flight conditions it is possible for a panel to be buckled due to constrained thermal expansion 
and also be subjected to very noise levels at the same time. An example is when a ASTOVL aircraft 
hovers close to the ground for an extended period, panels which are initially curved, or thermally 
buckled panels may possibly be snapped through from one side to the other by a large increase in 
dynamic pressure. 

“Snap-through” can be potentially damaging to the structure of an aircraft if it occurs persistently, 
because the process is associated with a large change in the cyclic bending stress present at the edge of 
a stiffened panel. High performance aircraft must therefore be designed so that snap-through never 
occurs in practice. 


1008 


The dynamic response of curved panels or buckled flat plates is difficult to predict theoretically because 
of non-linear effects. The established acoustic fatigue design techniques, which are based on linear 
vibration theory, are only able to provide approximate predictions of the loading regimes in which 
particular panels might be expected to undergo snap-through. 

The TDMC method can be used to model non-linear multi-modal vibrations of stiffened aircraft panels 
which are also subject to quasi-static stress. In particular calculations may be carried out in the post- 
buckling regime. 

With this technique simulated random dynamic pressure loading, with measured or otherwise known 
spectral characteristics is applied to a curved, or post-buckled panel and the time domain response 
calculated. The magnitude of the dynamic loading may then be increased until persistent snap-through 
is observed in the predicted response. This gives the designer the ability to design out the potential 
problem by systematically altering the most important parameters in order to identify the critical 
regime. 

Fatigue Life Estimation 

Although acoustic fatigue is a complex phenomenon, it has been established that the life of a component 
mainly depends on its stress/strain history. The most important factors in this regard are the magnitude 
and frequency of the cyclic strain and the mean level of stress at the likely failure points. On this basis 
fatigue life can be estimated by carrying out the three stage operation illustrated in Figure 1. 


Stage 1: Determine Loads 

A determination of static design loads is relatively straightforward compared to a calculation of th.e full 
temporal and spatial dependence of the aeroacoustic pressure on a military jet in flight. This is an 
enormous task in computational fluid dynamics (CFD). Designers have to rely on experimental data 
which can come from measurements on existing aircraft or from scale model tests of jets, for example. 
Existing databases can be extrapolated if the circumstances are similar. Experimental noise data is 
usually in the form of power spectral density curves as opposed to time series fluctuating pressures, but 
either can be used, depending on the circumstances. 

Stage 2: Calculate Stresses 

The technique chosen to obtain the stresses clearly depends on how much knowledge there is about the 
expected loads. In the early stages of design analytic techniques would be used to establish approximate 
sizes and stress levels. However, later on when the design is nearly completion, finite element (FE) 
stress analysis can be used to model the effect of random acoustic loading on the parts of the skin which 
are likely to be severely affected. These calculations would, of course, be done including the effect of 
attached substructure. 

The established method of designing against acoustic fatigue uses a frequency domain technique which 
relies on the validity of linear vibration theory. The method forms the basis of a number of methodology 
documents published by the Engineering Sciences Data Unit (ESDU). Whenever there^^e large out-of- 
plane deflections the frequency domain method cannot be used because of the “geometric non¬ 
linearity”. From a strictly theoretical point of view such analyses have to be carried out in the time 
domain, although approximate methods are applied with some success. 

The established technique produces inaccurate results for curved panels, buckled panels and for panels 
under high amplitude vibration. Geometric non-linearity usually stiffens a structure in bending so there 
is a tendency to overestimate the stress levels using the frequency domain technique. This conservatism 
is clearly useful from the point of view of safety, but it can lead to possible “over-design”. 
Unfortunately this is not always the case when there are compressive static stresses present. The 
established method also fails if the phase of the noise varies significantly over the surface of the 
structure, which is the case in a number of aeroacoustic problems. The techniques under development 
are designed to overcome these problems. 


1009 


Stage 3: Estimate Fatigue Life 


In cases of random acoustic loading it is customary to assume that damage accumulates according to the 
linear Miner's rule. Fatigue life is determined from experimental data in the form of stress (or strain 
amplitude), S, versus number of cycles to failure, N. If a number of cycles, n, of stress/strain, S, occur at 
a level of stress/strain where N(S) cycles would cause failure then the fractional damage done by the 
n(S) cycles is n(S)/N(S). 

Various methods have been developed for obtaining n(S) from the stress (strain) response. If the 
excitation is stationary, ergodic and the response is narrow band random then the function n(S) can be 
shown to be in the form of a Rayleigh distribution and the damage sum can be evaluated from plots of 
root mean square stress (or strain) against number of cycles to failure. If the statistics of the response 
are not Gaussian then it is necessary to count the numbers of stress cycles from the time domain 
response and use constant amplitude S/N curves. It is now widely accepted that the best way of 
counting the cycles is to use the Rainflow method, [1]. 

The Loading Regime 

The loads on an aircraft may be conveniently divided into static and dynamic. 

Loads which vary only slowly are: 

a) Steady Aerodynamic Pressure Loading, 

b) In-Plane Loads transferred from “external structure”, and 

c) Thermal Loads due to Constrained Expansions. 

The rapidly varying loads are, of course, the aeroacoustic pressure fluctuations which originate from 
any form of unstable gas or air flow. 

This division is central to the methodology which has been developed because it enables the modelling 
to be carried out in two distinct phases. The, so called, static loads do vary, of course, but the idea is to 
separate effects which occur on a time scale of seconds from the more rapidly varying acoustic 
phenomena. The aim is to split the loads so that the quasi-static effects can be calculated in an initial 
static analysis which does not depend on a particular dynamic loading regime. Any aerodynamic 
pressure may be divided into a steady part and a fluctuating part. The natural place to make the cut-off 
is at IHz which means that epoch times for TDMC simulations are then of the order of a second. The 
epoch time must not be too short because of statistical errors, and it cannot be too long because this 
would invalidate the assumption of constant quasi-static loads. In practice, there is another constraint on 
the epoch time. The number of finite elements in the model coupled with the premium on cpu time 
places an obvious limit on the epoch time. 

Comparison of the Time and Frequency Domain Methods 

A flow chart comparing the two methods is given as Figure 2. The main difference between the two 
techniques lies in the representation of the dynamic loads. The FD method uses rms loadings and 
spectral characteristics, whereas the TD method uses the full time series loadings. Gaussian statistics 
are, de facto, assumed by the FD method, but this is not necessarily the case with the TD technique. 

Application of the frequency domain method requires that the response is dominated either by a single 
mode or a small number of modes. To determine whether or not this is the case in practice, a normal 
modes analysis must be followed by a determination of the amount of coupling between the excitation 
and each mode. This can be determined quite accurately even if there is a certain amount of potential 
non-linearity by computing the joint acceptances for each mode, which are overlap integrals of the 
mode shape functions with the spatial characteristics of the excitation. Normally these quantities will be 
dominated by a few of the low order modes. If there is significant coupling into more than one mode 
then it will be necessary to use the TD method instead. 

With the time domain technique it is possible to represent the dynamic loads in a way which models the 
convection of the noise field across the structure. Very complicated loadings can be applied to large 


1010 


models but in consequence it can be difficult to validate the results obtained, because they cannot be 
checked against anything other than test data which is itself subject to confidence levels. In addition it 
must be remembered that the TDMC results themselves are subject to statistical variability. Finally it 
should be noted that TDMC data must be used in conjunction with constant amplitude endurance data. 
Rms fatigue data can only be used with frequency domain results. 

Time Domain Finite Element Modelling 

Until recently, the majority of finite element analyses were applied to static loading conditions or “low 
frequency normal modes analysis”. The method involves the use of an implicit code to invert in one 
operation, a single stiffness matrix, which can be very large. The general availability of extremely fast 
super-computers has now made it possible to carry out large scale non-linear dynamic finite element 
modelling using explicit FE codes. These codes use very similar types of element formulation to the 
implicit ones, e.g. shells, solids and bars, but the solution is advanced in time using a central difference 
scheme. 

One potentially very useful capability of time domain modelling is the application of acoustic pressure 
loadings which vary both in the time and spatial domains. If the spectral characteristics are known, 
either from test or from other modelling it is possible to generate samples of random acoustic noise and 
apply these directly to the finite element model as a series of “load curves”. 

The technique for determining time series noise was developed by Rice [2] and Shinozuka [3]. They 
showed that homogeneous Gaussian random noise can be generated from the power spectral density as 
a sum of cosine functions with different frequencies and random phase. Noise can be temporally and 
spatially correlated noise by deriving phase differences from cross spectral functions if they are known. 

The TDMC method can be quite costly in terms of central processor unit (cpu) time because the 
solution must be recalculated at each point in time. To reduce execution times, the explicit codes 
employ reduced numbers of volume integration points in the finite element formulations. However in 
this work cpu times are extended because long epoch times are required to ensure adequate statistics. It 
can take more than 24hrs to obtain a solution over a half second epoch if there are a few thousand 
elements in the model. 

Hence there is always a practical limit to the size of a particular time domain finite element analysis, 
(TDFEA). If the loading and geometry are not too complicated, the frequency domain method of 
analysis can be tried initially to gain more understanding of the nature of the response in an 
approximate way. In some cases the vibrational response regime must be considered carefully to decide 
whether TDFEA is really necessary. These may be situations where the non-linear effects are only 
moderate. 

It would be ideal if the full dynamic response of an aircraft could be determined with a fine mesh model 
in one huge operation, but experience has shown that this requires too many elements. It is possible to 
construct frill models with reduced stiffness using superelements, enabling flutter and buffet to be 
studied, because these are essentially low frequency phenomena. However, in time domain analysis it 
has been found that models containing a large amount of detail, such as fasteners and individual 
composite material plys, require a great deal of cpu time. To progress we must devise some strategies to 
overcome this situation. Since a full TDFEA can only handle a part of the aircraft structure, it is very 
important that loads external to the area under consideration are properly taken into account. This is 
cmcial to the success of this type of modelling as it is to all finite element modelling. 

The most important parameter in any time series analysis is the time step. This is determined by the 
velocity of sound waves in the structural material, and is generally of the order of a/v where a is the 
shortest element dimension and v is the velocity of longitudinal sound. A small time step is therefore 
required when the elements are small and the velocity of sound is large. For an aluminium model with 
10mm square elements the time step is about 1.6ixs. Hence a TDMC run with a half second epoch time 
needs about a half a million steps. A simple 5000 shell element calculation on a Cray C94 would take 
approximately 10 hours. 


1011 


Dynamic FE models of aircraft structure can be constructed in many ways, using shell elements, beam 
elements and/or solid elements. Special elements exist for damping and for sliding interfaces. Joints can 
be modelled with sliding interfaces, or with short beams, or just with tied nodes. Fasteners can be 
modelled with small solid elements, with short beams or with tied nodes, also. Unfortunately, however 
short beams and small solid elements cause a dramatic lowering of thC time step. For example, if the 
smallest fastener dimension is, say 3mm, the time step will have to be reduced to about O.Sfis if solid 
elements are used in the model. The effect on cpu time is such as to make the calculations impractical. 
Sliding interfaces are an efficient way to model skin/substructure contact in explicit analyses, but it is 
important to choose the algorithm carefully because some techniques can consume large amounts of cpu 
time. 

The best practical way of representing stiffened aircraft panels for TDMC analyses is considered to be 
with four noded shell elements simply tied together at their edges. A number of efficient shell 
formulations are avail-able and meshes can be rapidly produced from the design geometry. Of course, 
such models cannot be expected to produce highly accurate stress data in the region of small features 
but this aspect has to be sacrificed in the interests of achieving statistically significant amounts of time 
series data. To improve the accuracy of stress predictions in the neighbourhood of stiffeners etc., it will 
be necessary to couple TDMC analyses with fine mesh static analyses. 


The Generation of Time Series Data 

A number of factors must be borne in mind when generating time series data for TDMC calculations. It 
is important to consider carefully the frequency range and number of points which define the load 
spectrum in conjunction with the epoch time and number of points on the time series. 

The Nyquist Criterion [4] states that the time increment must be less than or equal to one over twice the 
upper frequency on the power spectral density curve. For the sake of argument, take the upper 
frequency to be IkHz. This means that the time increment must be less than 500ps. A more 
conservative time increment is based on the requirement to represent the dynamic response of the 
structure as accurately as possible over a full cycle. Assuming a resonant frequency of 500Hz, which is 
perhaps near the limit in practice, and 10 points per cycle which is more than sufficient, the lower limit 
on the time step works out at about 2C)0|is. Taking all these factors into consideration, the number of 
points on the spectrum curve should be of the order of 1000 and there should be between 1000 and 
5000 on the time series. Longer epoch times can be used but for reasons of practicality and statistics it 
is better to run more than one short epoch simulation rather than one long simulation. 

Explicit FE modelling frequently requires that the time step be smaller than 200jj,s. In the example 
given above the time step required by the explicit code was 1.6fis. Under these circumstances the 
random noise could be defined with a smaller time increment, but going to this level of effort has been 
found to produce no measurable change to the calculated response. 

Static Initialisation 

There are two possible ways of dealing with the effect of static loads in TDMC modelling. Firstly the 
complete analysis can be carried out using the explicit code. To do this it is necessary to apply only the 
static loads to the model and run the code until equilibrium is reached. By introducing a high level of 
artificial damping the stresses created can be relaxed in a relatively short period of time. The time 
required depends on the lowest resonant frequency of the structure and the size of the smallest element 
in the model. This facility is termed “dynamic relaxation”. 

The alternative is to make use of another facility in the explicit code called “static initialisation”. The 
deformed shape and stress state of the structure with just the static loads applied are first obtained very 
quickly using an implicit code. The solution for the stressed state is then initialised into the explicit 
code prior to the application of the dynamic loads. Dynamic relaxation may be used to smooth out any 
differences between the models. 


1012 



Damping Representation 

Vibrating aircraft structures are damped by several mechanisms, for example friction at joints, re¬ 
radiation of acoustic waves, and energy loss in viscoelastic materials It is difficult to generalise about 
the relative importance of each damping process in practice. Also reliable quantitative data is not 
available in sufficient detail to justify the inclusion of complex models of damping into the TDMC 
analyses. Test results on vibrating stiffened aluminium panels tend to show that the damping is best 
approximated by a combination of mass and stiffness proportional coefficients. There is a range of 
frequencies in which the damping ratio can be considered to be roughly constant. Until more detailed 
experimental data are available the most expedient approach is to assume a nominal value for the global 
damping ratio which does not change with frequency. Over the years it has become standard practice to 
assume a damping ratio of about 2% for fastened aluminium structures. 

Equivalent Linearisation 

There are some loading regimes in which the non-linear response to high levels of random acoustic 
loading can be approximately found using a linearisation technique combined with a frequency domain 
analysis. The basic idea is to replace the non-linear stiffness term in the general vibration equation by a 
linear term such that the difference between the rms response of the two equations is minimised with 
respect to a shifted fundamental resonance frequency. If an approximate equation for the non-linear 
stiffness is known then it is possible to derive an expression for the shifted “non-linear” resonance 
frequency. The rms response to random acoustic loading may then be found by combining the 
Miles/Clarkson equation with some form of static geometrically non-linear analysis. References to this 
technique are Blevins [5], Mei [6] and Roberts & Spanos [7]. Where the geometry is complex the most 
appropriate form of analysis is clearly finite element analysis. 

Implementation and Validation Studies 

The stress analysis work described in this paper has been undertaken using MSC-NASTRAN and 
LLNL-DYNA. 

NASTRAN is a well known implicit finite element code which is capable of handling very large 
numbers of elements. It has been developed very much with aerospace structural analysis in mind. It is 
basically a linear analysis code, although there are a large number of adaptations to deal with non-linear 
problems. It can also function as a dynamic code, but is much slower than DYNA in this mode because 
it basically needs to solve the complete problem at each time step. The non-linear features which are 
most relevant to the type of stress analysis being discussed here are those concerned with geometric 
non-linearity. Geometric non-linearity is treated by dividing the load into a series of steps, obtaining the 
solution incrementally. In this work, the code has been used for linear and non-linear static analyses and 
for normal modes analyses. 

DYNA is an explicit finite element code originally developed for the calculation of the non-linear 
transient response of three dimensional structures. The code has shell, beam and solid element models 
and there are a large number of non-linear and/or anisotropic material models available. DYTMA was 
developed primarily for the modelling of impact and there is no limit, as far as the code is concerned, 
on the size of finite element model which can be analysed. Many of its advanced features relate to 
impact modelling and are not required for this work. One useful feature, however is the laminated 
composite material model based on the equivalent single layer approximation. This code has been used 
for the TDMC calculations presented in this paper. 

Finite element models for the stress analyses were produced using MSC-PATRAN, It has a wide range 
of geometry and mesh generation tools and now has built in interfaces for both NASTRAN and DYNA. 
The element definitions are compatible with both codes and it is a simple matter to toggle between the 
two codes by changing the analysis preference. Not all the features of DYNA are supported and some 
of the parameters must be set by editing the DYNA bulk data produced by PATRAN. 

NASTRAN results were post-processed using PATRAN. DYNA results were post processed using 
TAURUS, which is faster and easier to use than PATRAN for this task; Some special in-house codes 
have been written to generate random acoustic noise from power spectral densities, as described above, 


1013 


and to post process time series output from TAURUS. One of the codes incorporates a fast Fourier 
transform (FTT) routine to determine spectral responses from the DYNA time series predictions. These 
codes are covered under the generic title “NEW-DYNAMIC”. 

TDMC Calculations on a Simple Flat Plate 

Calculations have been performed on a very simple model to implement the TDMC technique and 
develop the in-house software referred to above. A PATRAN database was constructed representing a 
simple flat plate, 350mm x 280mm x 1.2mm thick as an array of shell elements 34x28. For simplicity 
the boundary conditions were taken either as simply supported or clamped. There are a number of 
alternative shell element formulations available in DYNA, [8]. The Hughes-Liu shell was used initially 
because of its good reputation for accuracy, but later a switch was made to a similar, but slightly faster 
shell element, called the YASE. It was found that equally satisfactory results could be obtained more 
quickly using this element. 

Analyses without Static Loads 

A series of DYNA calculations were carried out with a fiiUy correlated random acoustic pressure load 
with a flat noise spectrum between OHz and 1024Hz. Investigations were carried out into the effect of 
varying the sound pressure level, the epoch time, the mesh resolution, the damping coefficient and the 
stochastic function. 

Figure 3 shows the displacement response of the central node of the model for a sound pressure level of 
115dB (about 12Pa rms), simply supported edges and mass proportional damping set so that the 
damping ratio was equal to 2% at the fundamental (1,1) resonance of the plate. The corresponding 
spectral response is shown in Figure 4. shows a sharp resonance peak at a frequency of 61.0Hz which is 
very close to the theoretical frequency of the (1,1) mode for the simply supported plate. The in-phase 
loading means that only the modes with odd numbered indices are excited. I^e peaks corresponding to 
the (3,1) and (1,3) modes are, however, not visible on the plot because they are too small. It may be 
concluded from these results that the behaviour of the plate at these pressure level is well within the 
linear regime. 

An investigation into the behaviour of the rms displacement response as a fijnction of SPL was carried 
out by increasing the loading incrementally from 75dB (0.12Pa rms) to 175dB (12kPa rms). The results 
are shown in Figure 5. Also shown are theoretical predictions obtained using the Miles/Clarkson 
formula with NASTRAN linear and non-linear analyses as explained above, see below for discussion. 

The statistical variation of the results was investigated by repeating a half second epoch TDMC run ten 
times with different samples of flat spectrum noise. It was found that the standard error of the rms 
response was about 16%. A second set of ten repeats were carried out with the epoch increased to 2.5s. 
In this case the standard error reduced to roughly 8%. From the theory of stochastic processes, it can be 
shown that the standard error is inversely proportional to the square root of the epoch time. On this 
basis therefore the ratio between the standard errors should be equal to the square root of five, or 2.23. 
From the analyses this ratio is about 2. Further runs established that these results are not affected by the 
vibration amplitude, even when the response becomes non-linear. 

Cautiously therefore, it can be concluded that the variance of the TDMC results is unaffected by non¬ 
linearity of the response. This is an important finding because it builds confidence in the technique. In 
many practical situations it may be necessary to rely on just one simulation and an appropriate factor of 
safety. It can be quite time consuming to carry out a large number of repeat TDMC simulations. The 
level of variance would be first established by repeating one load case a number of times, before 
confidently applying it to the results of other load cases. 

Comparisons with Linearised Theory 

The linear theory of plate bending, [9], leads to relationships between the central deflection, w, of a 
rectangular plate and a uniform static pressure load, Pstat which take the following form. 

Psutab = kcffW (1) 


1014 


where a and b are the length and breadth of the plate, and k^ff is an effective stiffness parameter which is 
a function of the modulus of rigidity of the plate and the edge boundary conditions. For the plate 
studied k^ff is about 30N/mm for the case of simply supported edges and lOON/mm for clamped edges. 
The above equation only holds, however, at very low amplitudes, as can be seen from Figure 6. This 
compares geometrically non-linear NASTRAN predictions with the linear ^eory. Curves are shown for 
both simply supported and clamped boundary conditions. The finite element results show the 
characteristic hardening spring type of non-linearity. 

At higher amplitudes the dynamic behaviour may be approximately predicted using “equivalent 
linearisation” theory, which assumes that the response remains predominantly single mode, but with a 
resonant frequency which rises as the stiffness of the structure increases. When the deflection is large 
the static force-deflection relationship can be written as the sum of a linear stiffness term and a cubic 
non-linear term: 

psiatab = kw(l+pw^) (2) 

where a b is the force, k is the linear stiffness. The equation is written with the leading term factored 
out to emphasis the point that p. is a constant which is small compared to the rms deflection. In the limit 
of small w we can expect the pw^ term in the brackets to be negligible compared to one, which means 
that the k in this equation must be the same as kcff above. 

Equation 2 was fitted to the NASTRAN results shown in Figure 6 to find the best fit values of k and p. 
Table 1 shows the results compared with the effective stiffness calculated from linear plate bending 
theory. It can be seen that the theoretical stiffness is almost identical to the best fit k from the non-linear 
finite element analysis. 

With reference to the results in Figure 5, it is obvious that the nature of the response is strongly 
dependent on the amplitude of the vibrations. For rms displacements up to about 4% of the plate 
thickness the behaviour was completely linear. For displacements between 4% and 150% of plate 
thickness, the response was essentially single mode dominated but the level could not be predicted by 
the Miles/Clarkson approach. The “equivalently linear” solution does, however, agree with the DYNA 
result up to a displacement of about l.8mm. The linearisation approach cannot be expected to be 
correct for displacements above about 1.5 times plate thickness. Above this point the response predicted 
by DYNA was multi-modal and strongly non-linear. The equivalently linear predictions departed 
considerably from the DYNA results when the vibration amplitude was very high. 

It was also observed that the frequency response peaks became increasingly noisy for higher pressures, 
representing the increased level of non-linearity in the plate vibrations. The increase in the frequency of 
the fundamental mode with acoustic pressure, as calculated by DYNA, is shown in Figure 7. Predictions 
from equivalent linearisation theory and from the theory of Duffing's equation are also included, see 
Nayfeh & Mook [10]. 


t„, = f(l + 3nw™,Y’ 

Equiv. Lin. Pred. 

(3) 

f„, = f(l+(3/8)w„,^) 

Duffing's Eq. Pred. 

(4) 


The DYNA results lie mostly between the two theoretical curves, agreeing particularly well with the 
results of equivalent linearisation theory up to around 700Pa (151dB rms). The level of agreernent 
obtained shows that the frequency response behaviour of the DYNA model is similar the theoretical 
predictions, providing an independent check on the results. As might be expected, at around 700Pa the 
agreement begins to breakdown, since the linearisation theories are not valid for deflections which are 
significantly greater than the plate thickness. It may be concluded, however, that the effect of geometric 
non-linearity at high amplitudes is being computed by DYNA in a reasonably accurate manner. A 
detailed comparison with experimental data is needed to determine the accuracy of the DYNA response 
predictions themselves. 


1015 


Due to the increasingly irregular shape of the frequency response functions derived from the DYNA 
time series predictions at high acoustic loads, it was not possible to calculate very accurate peak widths 
for pressures above approximately 135dB (I20Pa rmsX Figure 8 shows that the width of the peak 
increased with increasing acoustic pressure, but not in a regular manner. When the damping is mass 
proportional, equivalently linear theory predicts that the width of the peak should remain unchanged as 
the pressure rises. This is because the geometric stiffening effect of rising acoustic pressure exactly 
cancels the effect of a smaller damping ratio at the higher resonant frequency. This graph shows this as 
a horizontal straight line at 2.44Hz. The DYNA result is closer to the type of behaviour observed 
experimentally where the width of the peak generally increases with increasing the sound pressure level. 

Analyses with Combined Loads 

Further work was conducted with static loads superimposed on different levels of random acoustic 
loading. These calculations were done using the coupled NASTRAN-DYNA approach outlined earlier. 
That is to say the deformed geometry was obtained by applying the static loads to a NASTRAN model, 
with the results being initialised into DYNA and dynamically relaxed before the dynamic loading was 
applied. Calculations were performed with compressive in-plane loading, static pressure loading and 
thermal loading. With the exception of the thermal runs, the boundary conditions used in these runs 
were identical to clamped, except that symmetrical in-plane movement of the edges was permitted. We 
have called these conditions “semi-clamped”. It has been found that the fundamental resonant frequency 
of the plate without static loading is only reduced by a very small amount if the appropriate in-plane 
degrees of freedom are released, see Figure 9. These boundary conditions are actually closer to those 
which exist in reality when a panel in built into a larger structure. 

Figures 10-14 show results of some of the analyses which have been carried out. They give time series 
data along with spectra responses calculated by the in-house post-processing code. Numerical data 
derived from these results are summarised in Tables 3-5. 

A series of analyses have been carried out with compressive in-plane loads equal to one third of the 
theoretical buckling loads in compression. For the plate used, the forces per unit side length were - 
3.46N/mm in the x-direction and -5.46N/mm in the y-direction. The results of one analysis are shown in 
Figure 10. It has been found that the response remains dominated by the fundamental (1,1) mode as 
long as the plate is unbuckled and the SPL is low. The softening effect of the compressive loads on the 
frequency agreed quite well with Rayleigh-Ritz predictions, [5], up to an SPL of ?dB. At higher sound 
pressure levels, the DYNA results reflected stiffness changes which were greater than those predicted 
by the theory. The- same was found in the case of tensile loading. It is believed that these differences are 
due to approximations built into both the Rayleigh-Ritz theory and the DYNA code. 

Figure 11 shows the results of a calculation with a superimposed normal pressure. The magnitude of the 
pressure, 700Pa, was chosen so as to provide an example of “post-buckled” analysis. This size of 
pressure causes the plate to bow out in the centre by about 0.6mm. It is well known that in the post- 
buckling regime the random response of a plate depends upon the magnitudes of both the static and 
dynamic loads. In this case the static loading was large compared to the applied dynamic loads and 
“snap-through” did not occur. The plate simply vibrated about its statically deflected position in the 
fundamental mode with a slightly increased frequency. 

To provide a test of the DYNA thermal stressing capability, and to carry out an investigation into 
“snap-through”, several analyses were carried out with a uniform temperature rise of lOdeg C applied to 
the plate with clamped edges. This is quite sufficient to cause buckling because the resulting 
compressive biaxial stress, c, is well above the buckling level, Gb- If f is the frequency of the 
fundamental and J is a constant equal to 1.248 because of the clamped boundary condition, the two 
stresses can be determined approximately from 


a = EaT/(l-v^) 

(5) 

Cb = 4pa^f2/J 

(6) 


1016 


where E, a, v and p are Young's modulus, coefficient of thermal expansion, Poisson's ratio, and density 
respectively. Using these formulae we find c = 24MPa and Ob = 14MPa. 

Analyses were carried out with several different levels of dynamic load. The results of three of the 
calculations are shown in Figures 12,13 and 14. It was found that the threshold for snap-through 
occurred at an acoustic load of about IkPa, see Table 4. Below this level the mean deflection, w, is a 
function of the static load alone, equal to about 2.8nun (the negative sign indicates that the plate has 
bowed in direction of negative z). At higher SPLs the mean deflection reduces because the plate snaps 
backwards and forwards between positive and negative z. The calculated response spectra for these 
higher level runs, show an additional peak at a very low frequency, ie less than lOHz. This is an artifact 
caused by the snap-through since the fundamental resonance of the clamped plate is at 113Hz. 

Figures 10-14 all show probability density functions derived from the time series data. The fluctuations 
on these plots are caused by the smallness of the epoch time. In all cases, except for the thermal 
calculations with the two largest acoustic loads, it can be seen that the functions are basically Gaussian 
in shape. It may therefore be concluded that it is reasonable to assume that the response of a plate in the 
post-buckled region is Gaussian unless there is a large amount of snap-through. 

Discussion 

The work described is the starting point for investigations and validations using more complex FE 
models. Further work has been carried out using models including curvature, sub-structure and detailed 
features. It is difficult to validate the predictions obtained from such models by comparing with test data 
because the results themselves are open to interpretation. It has been found that the predicted stress 
levels are closer to the test results when the chosen location is away from any small features. The lack 
of good agreement in the neighbourhood of the features can be explained by the relatively coarse mesh 
used in the dynamic models. The overall level agreement was much better than that between predictions 
based on linear or equivalently linear theory and test. On the basis of experience, the latter tend to over¬ 
predict by upwards of factors of two and three. From this work it has been found that the DYNA 
predictions tend to be greater than test by amounts which vary but are generally much less. The average 
over-prediction was about 40% with a significant change as a function of location. 

TDMC runs can take a significant amount of computer time to carry out and it is believed that to make 
further improvements the technique should be combined with detailed stressing using static finite 
element analysis. Inaccurate results can be obtained if the boundaries of the part of the structure under 
analysis are not properly restrained. In the case of models of aircraft panels this may significantly affect 
the resonant frequencies which in turn affects the level of calculated dynamic stress. In-plane loads on a 
panel, perhaps due to thermal stressing, can alter the fundamental by as much as 100-200Hz. Looked at 
from a theoretical point of view, the only way to solve this problem is to construct a second, coarse 
model of the component, along with some of its surrounding structure. An initial calculation can then be 
carried out with this model in order to obtain the loads and boundary conditions for subsequent 
application to the original model. 

The dynamic phenomenon of “snap-through” cannot be modelled using existing methods and so the 
TDMC / finite element technique offers the engineer a way to determine where the likely regions of 
unstable vibration are located in circumstances where the structure is complicated by attachments etc. 


Conclusions 

This paper has sought to explain how time domain finite element modelling can be used to assist in the 
design of aircraft against acoustic fatigue. Although the technique is computationally intensive, it does 
have a place in the effort to understand complex vibrations, such as the response of structures to 
spatially correlated jet noise excitations, or interactions between high sound pressure levels and thermal 
loads. 

The work at BAe is continuing in an attempt to provide the analyst with a greater ability to determine 
dynamic stress levels in advanced structures with complex loadings. 


1017 


References 


1. Dowling N. E. , Fatigue Prediction for Complicated Stress Strain Histories, J Materials 1, 71 
(1972). 


2. Rice, In Selected Papers on Noise and Stochastic Processes, Ed N Wax pplSO, Dover New York 
(1954). 

3. Shinozuka M. , Computers and Structures, 2, 855, (1972). 

4. Bendat J. S. and Piersol A.G. , Engineering Applications of Correlation and Spectral Analysis 
Wiley (1990). 

5. Blevins R.D, , An approximate method for sonic fatigue analysis of plates and shells, J Sound and 
Vibration, 129, 1, 51 (1989). 

6. Mei C. and Paul D.B. , Non Linear multi-modal response of a clamped rectangular plates to 
Acoustic Loading, AIAA Journal, 24, 634, (1986). 

7. Roberts J. B. & Spanos P.D , Random Vibration and Statistical Linearisation, Wiley, (1990). 

8. Whirley R.G. and Engelmann B.E. , DYNA3D: A Nonlinear, Explicit, Three Dimensional Finite 
Element Code for Solid and Structural Mechanics — User Manual, Lawrence Livermore National 
Laboratory, UCRL-MA-107254 Rev. 1, (Nov 1993). 

9. Szilard R., Theory and Analysis of Plates, Prentice Hall, New Jersey. 

10. Nayfeh A.H. & Mook D.T. , Non Linear Oscillations, Wiley (1979). 


1018 



Parameter 



lUQgnnggll 




HISSH 


Linear Theory 

NASTRAN 

NASTRAN 

Simply Supported 

30.0 

30.7 

1.09 

Clamped 

104. 

101. 

0.266 


Table 1: Values of Parameters fitted to NASTRAN results compared with the 
linear theory values. 


Static Loads 

Acoustic Load 
Rrms 
(Pa/dB) 

Theory ] 

1 DYNA 

(N.mm 

N, 

(N.mm 

Freq 

(Hu) 

Wfim 

(mm) 

■SI 

Wrms 

(mm) 


None 

1.2 (95.6) 

jnggi 

0.00290 

114 

0.01.2 


-3.46 

1.2 (95.6) 


0.00316 

94.1 

0.00313 


-3.46 

643.5 (150.1) 


1.69 

115 

1.08 


-3.46 

1.2 (95.6) 

68.33 

0.00359 

68.1 

0.00368 


-3.46 

700 (150.9) 

68.33 

2.09 

103 

1.42 


Table 2: Summary of results of calculations with random acoustic loading 
superimposed on compressive in-plane loads. 


Static Load 
p(Pa) 

Acoustic Load 
(Pa/dB) 

DYNA 




None 

1.2 (95.6) 

114 

0.0000 

0.01.2 

700 

None 

— 

-0.646 

— 

700 

12 (115.6) 

115 

-0.655 

0.0189 

700 

700 (150.9) 

134 

-0.477 

1.28 

5 k 

None 

— 

-2.89 

— 

5 k 

12 (115.6) 

179 

-2.93 

0.0152 

5 k 

700 (150.9) 

173 

-2.70 

0.999 


Table 3: Summary of results of calculations with random acoustic loading 
superimposed on static pressure loads. 


Temperature 
T (<»C) 

Acoustic Load 
(Pa / dB) 

1 DYNA 1 




None 

1.2 (95.6) 

114 


0.00256 

10 

None 

— 

-2.85 

— 

10 

1.2 (95.6) 

234 

-2.86 

0.0140 

10 

700 (150.9) 

219 

-2.80 

0.197 

10 

Ik (154.0) 

9.01 

2.47 

1.02 

10 

1.2k (155.6) 

5.01 

-0.717 

2.24 

10 

1.5k (157.5) 

3.00 

-0,376 

2.20 

10 

2k (160,0) 

92.2 

0.0185 

1.95 

10 

4k (166.0) 

195 

0.0568 

2.08 


Table 4: Summary of results of calculations with random acoustic loading 
superimposed on a thermal load. 


1019 































Figure 2: Flowchart Illustrating the Frequency and Time Domain Techniques 



(uiUi),ueujeoB,ds,a ^ O 



Displacement of central node for 12Pa rms acoustic pressure, 

1 . 



DYNA 


Figure 4: Spectral Response of the fiat plate corresponding to Fig 3. Central 
Node for 12Pa rms pressure, DYNA calculation. 









3 


2.5 


2 

1 

o 

^ 1 


0.5 


0 

Figure 5: Rms Central Deflection of the plate versus sound pressure level 
Comparison between DYNA results and linear theory. 


3 


2.5 


2 

■£ 

£ 

E 

I 

I 

o 

1 


0.5 


0 

Figure 6: Central deflection of the plate versus pressure, NASTRAN calcula¬ 
tions compared to linear theory. 



500 1000 1500 2000 2500 3000 3500 

Static Pressure (Pa) 



rms Pressure (Pa) 


1022 





Figure 7: Variation of fundamental frequency of the (1,1) mode with rms 
pressure, Comparison between DYNA and theory. 



Damping 2%; Stoctiastic Fn #1; 3Sx29 Nodes 

0 - 1 - 1 - 1 - 1 -- 

0 200 400 600 800 1000 1200 

rms Pressure (Pa) 

130 140 145 150 153 155 

SPL (dB) 

Figure 8: Variation of width of the (1,1) mode resonance peak with rms pres¬ 
sure, Comparison between DYNA and theory. 


1023 





Figure 9: DYNA model predictions for random vibration of the plate with 
semi-clamped boundary conditions. 



Figure 10: Random Vibration results with compressive load in the y-direction 
of -3.46N/mm and SPL of l.OPa. 



Figure 11: Random Acoustic Loading of SPL=12Pa superimposed on a static 
pressure of TOOPa. 


1024 










Figure 12: Random acoustic loading of SPL=700Pa superimposed on a thermal 
load of 10 deg, clamped edges. — No Snap Through 



Figure 13: Random acoustic loading of SPL—2kPa superimposed on a thermal 
load of 10 deg, clamped edges. —Nearly continuous snap-through. 



Figure 14: Random acoustic loading of SPL=4kPa superimposed on a thermal 
load of 10 deg, clamped edges. —Dominant acoustic load. 
















1026 


SYSTEM IDENTIFICATION II 


ROBUST SUBSYSTEM ESTIMATION 
USING ARMA-MODELLING 
IN THE FREQUENCY DOMAIN 

by U. Prells, A. W. Lees, M. 1. Friswell and M. G. Smart, 

Department of Mechanical Engineering of the University of Wales 

Swansea, 

Singleton Park, Swansea SA2 8PP, United Kingdom 

ABSTRACT 

This paper reflects early results of the research on modelling the influence of 
the foundation on the dynamics of the rotor. The foundation is connected to 
the rotor via journal bearings. Dynamic models exist for the subsystems of 
the rotor and of the bearings; the first is reliable but the latter is uncertain. 
The foundation model is unknown and has to be estimated using rundown 
data.These are measured responses of the foundation at the bearings due 
to unbalance forces of the rotor which are assumed to be known. Uncer¬ 
tainties in the bearing model will be transfered to the estimated foundation 
parameters. The main scope of this paper is to introduce a method which 
enables the decoupling of the problem of model estimation and the problem 
of the influence of the bearing model uncertainty. 

The influence of changes in the model of the bearings on the estimation 
of the foundation model is mainly due to the sensitivity of the computed 
forces applied to the foundation at the bearings. These are used together 
with the associated measured responses to estimate the foundation model in 
the frequency domain. Using an ARMA model in the frequency domain it is 
possible to estimate a filtered foundation model rather than the foundation 
model itself. The filter is defined in such a way that the resulting force has 
minimum sensitivity with respect to deviations in the model of the bearings. 
This leads to a robust estimation of the filtered model of the foundation. 
Since the filter can be defined in terms of the models of the rotor and of the 
bearings only, the problems of estimating the foundation’s influence and of 
the sensitivity of the estimates with respect to the model of the bearings 
are decoupled. 

The method is demonstrated by a simple example of a single-shaft rotor. 
Even if the errors in the bearing model are about 50 % the relative input and 
output errors of the filtered foundation model are of the same magnitude 
as the round-off and truncating errors. 

1 INTRODUCTION 

An important part of a machine monitoring system for fault diagnostics of 


1027 


a turbo generator is a reliable mathematical model. This model includes 
the subsystems of the rotor, the bearings and the foundation. The model of 
the rotor represents the most reliable knowledge, the model of the journal 
bearings is uncertain, and despite of intensive research it is not yet possible 
to define a model for the foundation which refiects the dynamical contribu¬ 
tion to the rotor with sufficient accuracy. The first step to determine the 
contribution of the foundation on the rotor’s dynamic performance is the 
estimation of a reliable foundation model. 

Rundown data are available, i.e. displacements ufb{<->^) ^ of the 
foundation at the bearings which are due to an unbalance force /c/(a;) G 
of the m-shaft rotor, given at discrete frequencies a; € := {wi, ■ • •, wm}? 

and this data may be used to estimate the foundation model. A com¬ 
mon method ([1],[2],[3],[4],[5]) is to estimate the unknown dynamic stiffness 
matrix 6 i^4mx4m foundation at the bearings using the in¬ 

put/output equation 

F{uj)upb{(^) = (1) 

where the force fpB of the foundation at the bearings can be expressed 
by dynamic condensation in terms of the data ups, fu and in terms of the 
dynamic stiffness matrices Ar,B of the rotor and the bearings respectively 
yielding 

fpB = -Bufb + [ 0 , B]A~^q ^ ^ . ( 2 ) 

Here the dynamic stiffness matrix Arb of the rotor mounted on the bearings 
is partitioned with respect to the n inner degrees-of-freedom (dof) of the 
rotor and to the 4m connecting (interface) dof 


^RB 


Aru Arib 
Arbi Arbb + B 




(3) 


The non-zero components of the force fpi G (D” in eq. (2) of the inner part 
of the rotor are the components of the unbalance force /y, i.e. introducing 
the control matrix Su € ^ dynamic stiffness 

matrix Ar of the rotor is given in terms of the matrices of inertia and 
stiffness which are defined by modal analysis and by its physical data given 
by the manufacturer. Each of the m shafts of the rotor is connected to 
the foundation usually via 2 journal bearings. Since the dynamic stiffness 
matrix B of the journal bearings represents a model for the oil film it consists 
of connecting dof only. It can be shown that B is block diagonal 




B = 


0 


0 




(4) 


containing the dynamic stiffness matrices Bi = Ki+ jujDi.i = 1, • • • ,m, of 
the m bearings. Ki, Di are the matrices of stiffness and damping respectively 


1028 


which result from linearisation and are in general non-symmetric and non¬ 
singular. Eq. (1) is then used to estimate the foundation transfer function 
F{u). This has been discussed in several papers ([1],[2],[3],[4],[5]). Lees et 
al. [3] pointed out that fpB is sensitive with respect to deviations in the 
model of the bearings over part of the frequency range. This sensitivity is 
transfered to the model estimates. 

In this paper a method is introduced which enables the decoupling of the 
two problems of model estimation and of sensitivity of the foundation model 
with respect to the model of the bearings. The basic idea of this method is 
to estimate a transfer function H{u) which maps the displacements ufb{^) 
to a force /^(w) rather than the force i-e. 

H(u)ufb{^) = ( 5 ) 

In extension of the earlier method the force //f(a;) can be chosen to be of 
minimum sensitivity with respect to the model of the bearings. This robust 
estimated transfer function H{uj) is related to that of the foundation F(uj) 
by a transformation P{u) 


H{lo) = PMF(a;), (6) 

which of course retains the sensitivity with respect to the model of the 
bearings. But since P{uj) only depends on the models of the rotor and the 
bearings in the case of a modification within the model of the bearings no 
new model estimation has to be performed because this has been done ro¬ 
bustly with respect to such model changes. 


2 THE OPTIMUM CHOICE OF THE 
FORCE VECTOR 

As stated in Lees et al. [3] the sensitivity of the force /fb with respect to B 
is mainly due to the inversion of the matrix Arb in eq. (2). It can be shown 
that the condensation method of estimating the force /fb results from the 
special case of eliminating the last 4m rows of the matrix 


W := 


Arii 

^RBI 

0 


Arib 

ArbB -b P 


^ ^(n+8m.)x(n+4m) 


( 7 ) 


which can be written as Arb = T'^W ^ (l^(n+4m.)x(n+4m) defining the 
selecting matrix of the master dof as 

T := [ei, • • • , € R(-+8n)x(n+4m)^ (S) 


1029 


where in general en denotes a unit vector of appropriate dimension contain¬ 
ing zeros everywhere but in the nth place. In extension to the force fpB 
defined in eq. (2) for an arbitrary selecting matrix T € the 

condensation leads to a force fn given by 




j-^LT 


In+Sm-W{T'^Wy'T^ 

” *■ V *' .. . 

=:P 


( 

Bufb 

V } 


( ^ \ 
0 

V -^4m J 


fpB- 


( 9 ) 


Here T-^ € denotes the matrix which selects the slave dof, 

and in general In denotes the unit matrix of dimension n. Indeed, inserting 
the special choice of T from eq. (8) into eq. (9) leads to the sensitive force 
Ih = fpB as defined in eq, (2). 

The reason for the sensitivity of /fb is that the subsystem of the rotor 
has low damping. Near the resonance frequencies of Arb its large condition 
number depends sensitively on B. Thus the sensitivity of Jfb with respect 
to B is due to a large condition number of Arb • Let T denote the set of all 
possible selecting matrices, i.e. 

r := {[ei..--',ew4„]:e4elR"+*“. 

l<4<n + 8m, V /c = 1, • •«,n +4m}, (10) 

One criterion for an optimum choice of the force Jr may be formulated as 
the following minimisation problem: 

Criterion 1: 

The optimum choice is the solution of 

mmcond{T^W), (11) 

where W is defined in eg. (7). 

A low condition number is necessary but not sufficient in order to provide 
a low sensitivity of the force /h* Therefore a numerical test can be applied 
using stochastic deviations in the bearing model. Let A A consist of 

uniform distributed non-correlated random numbers with zero mean values 
and variances equal to 1/3 for alH G 1, ■ • *, 2m. Define 

ABi = ABfisi, Ti) := siAKi -h A A, (12) 

where the positive scalars si.Ti control the magnitude of the random error 
of the i-th bearing model. Thus, the error AB = AB{s,r) of the bear¬ 
ing model is well defined for s := (si, • ■ •, 52771 )"^ and r := (ri, • • • ,r 2 mV■ 


1030 


Regard the force fn = f{uj,T,AB) as a function of the selecting ma¬ 
trix T and the bearing model error AB. For I random samples AB{k) = 
AB(r{k),s{k)),k == calculate for each frequency a; G the up¬ 

per and lower bounds for the real and imaginary part of each component 
/i, ^ = 1, • • •, 4m of the force vector /, i.e.: 


//Lax(“.U 

;= max Re {/i(a;,T, AB(/c))} , 

(13) 


:= min Re {fi{u,T, AB{k))} , 

(14) 


:= ^max^Im {/i(a;,T, AB(/c))} , 

(15) 

fLni^.T) 

:= min Im AB{k))} . 

k=l,"-,l 

(16) 

Defining the force vectors 



/max(w,T) 

■■= T)+j- T), 

(17) 

/mm(w,T) 


(18) 


the second criterion can be formulated as a minimax problem: 

Criterion 2: 

The optimum selection is obtained from 

4m 

minmax^l/imax(w,r) - (19) 

TST uj£u . - 
2 = 1 

Before the method outlined is demonstrated by an example some aspects of 
the mathematical model of the foundation and methods for its estimation 
based on the input/output equation (5) will now be considered. 


3 ESTIMATION OF THE FOUNDATION 

MODEL 


The purpose of this section is to estimate the unknown foundation model 
represented by the matrix 

F{uj) = Afbb{^) - Afbi{^)Af]j{uj)Afib{^)‘ (20) 


This expression results from dynamic condensation of the dynamic stiffness 
matrix of the foundation 


Af 


Afbb Afib 
Afib Afh 


( 21 ) 


1031 



which is partitioned with respect to its inner dof (index I) and those dof 
coupled to the bearings (index B). For viscous damped linear elastomechan- 
ical models the dynamic stiffness matrix Aj?(a;) of the foundation is given 
by ^ 

Af(uj) := ( 22 ) 

i=0 

The matrices Ai are real valued and represent the contributions of stijffness, 
damping and inertia for z = 0,1,2 respectively. In this case the identifica¬ 
tion of the foundation model requires the estimation of the three matrices 
Ai which are parameterised by introducing dimensionless adjustment pa¬ 
rameters aik € IR, for all /c = 1, * • •, ^ = 0,1,2 (see for instance [6] or 

[7] ). Those parameters are related to given real-valued matrices Sik by 

Ni 

Ai{ai) 

Jk=i 

Writing the adjustment parameters as one vector a"’’ ;= (af,aj’,aj) G IR^, 
p := ATq + Ni + W 2 , the estimation of the foundation model is equivalent to 
the estimation of the parameter vector a. The dynamic stiffness matrix of 
the foundation becomes a nonlinear function of this parameter vector 

^"( 0 ;) =F[(jj^a) — Afbb[^-)0) “ Afb7(ci;j a)Ap}j(a;, a)Af’/5(a;, a). (24) 

Substituting the measured quantities for upB and fu into eqs. (5) and 
(9) the parameter vector a is usually estimated by minimising some norm of 
the difference between measured and calculated quantities, called residuals 

[8] . Using equation (5) is equivalent to the input residual method. Defining 
the ith partial input residual as 

(25) 

where the dependency on the model parameters a of the input vector is 
defined by 

f{u), a) := P{uj)F[u), a)uFB(w), (26) 

the cost function to be minimised is given by 

M 

JKa):=£^;}(i)W,(i)^/(i), (27) 

i=l 

where Wj{i) represents a weighting matrix for the zth partial residual and 
the superscript f denotes the conjugate-transpose. The inverse problem (27) 
is nonlinear with respect to the parameters to be estimated. Thus, there 
is no advantage relative to the output residual method. Defining the ith 
partial output residual as 

vo{i) := u{ui, a) - (28) 


1032 


where the model output is defined by 


(29) 


0 

0 

ApiiiuJ^a) . 
(30) 


(31) 

Woii) denotes a weighting matrix for the ith partial output residual. 

Mathematical modelling is always purpose orientated [9]. In the case 
discussed in this paper the purpose is to estimate the influence of the foun¬ 
dation on the dynamics of the rotor. For this purpose, no physically inter¬ 
pretable model is necessary in order to model this influence. In the next 
section an alternative mathematical model is introduced which leads to a 
linear inverse problem. 

3.1 ARMA MODELLING IN THE FREQUENCY 
DOMAIN: THE FILTER MODEL 

Auto Regressive Moving Average models are well developed (see for in¬ 
stance [10],[7]) in order to simulate dynamic system behaviour. ARMA 
models are defined in the time domain by 

- lAt) = - ^A^), (32) 

i=0 i=0 

where the present output (state or displacement) u{t) due to the present 
input f{t) depends on rio past outputs and on rii past inputs. 

In the frequency domain eq. (32) leads to a (frequency-) filter model [7]. 
With reference to eq. (5) it has the form 


Su 

0 

0 

0 


u(w,a) := [0,0,/2(m+i),0].4 ^(w,a) 
with the dynamic stiffness matrix A of the entire model 
A{u, a) = 




Arii{u) Arib{^^) 0 

Arbi{^) Arbb{‘^)B{lo) —B(u) 

0 —B{uj) R(a;) 4 -Afbs(^) g) 

0 0 Afib{^jO,) 


the cost function to be minimised is 


M 


Jo(a) :=E^oW^oW«o(i). 


The output and input powers Uo^rii respectively, and the matrices 
(•SA:)fc=o,-,ni are called filter^parameters and Jiave to be esti¬ 
mated. Of course the minimum of det[A(a;)] and of det[B(cj)] correspond 
to the resonance and anti-resonance frequencies of the subsystem of the 
foundation respectively. 

For an optimum choice of P (see eq. (9)) the estimation of A and B can 
be considered to be independent of the precise values of the model of the 
bearings. Thus, the problem of the uncertainty in the bearmg models and 
the problem of model estimation are decoupled. If A and B are estimated 
refering to eq. (5) then 

= H. (34) 

The estimation of the filter parameters is robust with respect to deviations 
in the bearings model. Thus, the uncertainty of the estimation of the foun¬ 
dation model F is due to the inversion of the matrix P only 

F = P-^H = p-'-B~^A, (35) 

which represents a problem a priori and which occurs only in the calculation 
of the force of the foundation at the bearings 

/PB = P-^B~^Aufb- (36) 

Of course the force vector fpB is sensitive to changes in the bearings model 
but only due to corresponding changes in P. The estimated part B A is 
robust with respect to changes in the bearings model. 

In order to calculate the response ufb iio explicit calculation of the 
inverse of P is necessary, 

upB = A Bfn- (37) 

Since the estimated model and the force Jh are insensitive with respect to 
the bearings model the estimation otupB is robust in this sense. 

Of course the influence of errors in up b and fu have not yet been taken 
into account. Accordingly the model powers n^n^, must be estimated as 
well as the matrices Ai.Bk- The estimation method is outlined in the fol¬ 
lowing section. 

3.2 ESTIMATION OF THE FILTER 
PARAMETERS 

In order to estimate the filter parameters the least squares method can be 
applied to minimise the equation error in eq. (33). Defining the zth partial 
equation residual as 

vsij) '= A[ui)upB[^i) - B{uJi)fH{^i) (38) 


1034 


the cost function to be minimized is given by 


M 

i=l 


(39) 


where W^ii) denotes a weighting matrix for the ith. equation residual. As¬ 
suming WeIi) = Am for alH = 1, • • ’, M, the filter equation (33) can be 
extended for M excitation frequencies as 

= (40) 


where C/, 2^ and A are defined by 


U 

Z 

A 


• • • 5 , 

r a;i 0 1 



lom 


(41) 

(42) 

(43) 


The solution of the minimisation problem (39) is equivalent to the normal 
solution of eq. (40) which can be rewritten as 


[Afio 1 ''' 1 -Ao ) ‘ : ■^o] 


=: V 


U 

Z 


= 0 


(44) 


Because the filter parameters represented by the matrix V € 
j^ 4 mx 4 m(no+ni+ 2 ) leal-valued, equation (44) must be satisfied for the 

real and imaginary parts of the matrix Y € which finally 

yields 

1/[Re {Y} , Im {Y}] =: VA = 0. (45) 

This problem does not lead to a unique solution for the filter parameters. 
Indeed, for any arbitrary non-singular matrix C 

CAufb = (46) 


is also a solution. But since one is interested (see eq. (34)) in the product 
B~^ A (or its inverse) only this final result is of interest and this product is 
unique. 


1035 


As a necessary and a sufficient condition for a full-rank solution V of 
eq. (45) the matrix X € 5 ^ 4 m(no+ni 4 - 2 )x 2 M a rank deficiency of 

4m, i.e. 

rank(A) = 4m (no + n^ + 1 ). (47) 

Of course this problem has to be treated numerically. The rank decision 
is usually made by looking to the singular values 7 (no, rii) 6 5^4m(no+ni+2) 
of the matrix X = X{no,ni). Because one cannot expect to achieve zero 
rather than relative small singular values one has to define a cut-off limit. 
This is due to the fact that the equation error (39) can be made arbitrary 
small by increasing the degree p := no + Ui of the filter model. The same 
situation occurs if one looks at the maximum relative input error 


6 / := 


max 




(48) 


or to the maximum relative output error 


en :== max 


II^fs(^2)1I 


(49) 


With increasing degree p the errors ej and cq can be made arbitrary small. 

This is a typical expression for an ill-posed problem which can be turned 
into an well-posed problem by applying regularisation methods [ 11 ]. To 
choose an appropriate regularisation method needs further investigation and 
is beyond the scope of this paper. In the next section the method of choosing 
the optimum force vector fn is demonstrated by a simple example. 


4 A SIMPLE EXAMPLE 

The test model is depicted in Fig. 1. The one-shaft-rotor is simulated by 
an Euler-Bernoulli beam which is spatially discretised with 10 dof. Accord¬ 
ing to the partition with respect to inner points and interface points (see 
eq. ( 3 )) the number n of inner rotor dof is 8 and the number of connecting 
dof is 2 . Only one translation dof of the rotor is connected to each bearing 
which are modelled by massless springs with stiffnesses ki — 1.77 • 10® and 
k 2 = 3-54 • 10® N/m respectively. The foundation is modelled by an uncon¬ 
nected pair of masses mi = 90, m 2 = 135 kg and springs with stiffnesses 
kfi = /c /2 = 1.77 • 10® N/m. The force fu due to an unbalance 6 = 0.01 
kg-m is given by fu{<^) ■= € IR. The force vector f^i e IR® is assumed 

to have one non-zero component only, i.e. Jri fuc^. The frequency range 
between 0 and 250 Hz is discretised with equally spaced stepsize of 0.5 Hz. 
The selecting matrix T € of the master dof is assumed to consist of 

the unit vector 64 in order to select out the unbalance force fjj because this 


1036 



Figure 1: The simple test model 

excitation is independent of the model of the bearings and therefore of min¬ 
imum sensitivity. Thus the remaining redundancy consists in eliminating 
one row of the matrix 

G := [ei, 82 .83. 65. • • •. eul'' W € (50) 

The result of the first criterion are depicted in Fig. 2 . It shows the frequency 
dependent condition numbers for the elimination of each row of the matrix 
G in turn. This leads to an optimum choice by eliminating the 4-th row of 
G. Thus, the optimum choice of the master dof is given by T*^ = [ 64 , 65 ]. 
This result is confirmed by applying the second optimisation criterion. 

For this purpose a uniform distributed uncorrelated random error with 
zero mean value is added to the stiffnesses of the bearings simulating a 
model variation of 50 per cent, i.e. ki —^ ki -1- Nzikil2, where = 

1 , 2 , are uncorrelated random numbers with expectation value E{Aki} = 0 
and with variance E{AkiAkj} = For a size of I - 500 random 

samples, the upper and lower bounds /max,/min and /f max,/ h min of the 
force fpB = /fb(<^, A5) and of the force Jh = /(w,T, AB) respectively 
with the selecting matrix T - [ 64 , 65 ], have been calculated. In contrast to 
the maximum difference of upper and lower bounds of the force f^B of ~ 20 , 
that of the force fn is of the order of the computational accuracy ~ 10 "'^, 
and is therefore negligible. In a first step the force /fb is used to estimate 
the filter parameters of the model F of the foundation. Solving the singular 
value decomposition for all input and output powers {ni.rio) 6 [0, 5]^ C 
the calculation of the maximum relative input and of the maximum output 
errors as defined by eqs. (48) and (49) with fn ~ Ifb leads to the results 


1037 




Elimination of row number... 



Frequency [Hz] 



Frequency [Hz] 

Figure 2: Frequency dependent condition numbers 

depicted in Fig. 3. For a model realisation with [ni.rio) = (0,2) the values 
of the maximum relative errors are approximately ej 7.1 • and eo ~ 
9.0-10“”^ which corresponds to the computational accuracy. Using this model 
the estimates of the filter matrices correspond within the computational 
accuracy to those of the ‘true’ foundation model. 

Using variations of the force fpB between the bounds /max, /min froni 
the second criterion the associated upper and lower bounds of the relative 
input and output error have been calculated. The influence of the variation 


1038 













Firgure 3: Maximum relative input and output error for different 
input powers rii and output powers rio using the force vector fpB 


of the bearing stiffness of 50% leads via the associated variation of the 
force fpB to drastic variations of the relative input and output errors. The 
difference of upper and lower bound of the maximum relative input error is 


1039 







Output power 


Figure 4: Maximum relative input and output error for different 
input powers rii and output powers Uo using the optimised force 

vector Jh 

of order 100 and that of the maximum relative output error is approximately 
LOl 







The situation is different using the optimised force fn in order to es¬ 
timate the filter parameters of the model H. For each input and output 
power {ni,no) e [0,25]^ C the maximum relative input error and the 
maximum relative output error have been calculated. The result is shown in 
Fig. 4. For a maximum relative input error e; 6.5 • 10“® a filter model of 
degree 12 is available with the powers (n^, = (4,8). This model produces 

a maximum relative output error eo ^ 10“®. 

Analogous to the robustness investigations for fpB now for the force 
/h the upper and lower bounds of the relative input and of the relative 
output error due to the random variation in the bearing models have been 
calculated. For the chosen model with powers (nj,no) = (4,8) the difference 
of upper and lower bound of the relative input error as well that of the 
relative output error are of about the same order 10“^. Thus, compared 
with the order of variation 100 and 10"^ of the direct foundation model 
estimate the estimate of H is robust with respect to changes in the bearing 
model. 


5 CONCLUSION 

In this paper a method is introduced which enables the decoupling of the 
two problems of model estimation and of sensitivity of the foundation model 
with respect to the model of the bearings. The method produces an opti¬ 
mised choice of the input/output equation which provides a transfer func¬ 
tion estimation that is robust with respect to deviations in the model of 
the bearings. For the foundation model estimation a filter model is intro¬ 
duced. This modelling strategy has the advantage of leading to a linear 
inverse problem. The disadvantage is that with increasing model degree 
the equation error can be made arbitrarily small. Because this error should 
not become smaller than the accuracy of the data, a cut-off limit has to be 
determined a priori. Further investigations should allow the cut-off limit to 
be related to the data errors. 

REFERENCES 

1. Feng, N.S. and Hahn, E.J., Including Foundation Effects on the Vi¬ 
bration Behaviour of Rotating Machinery. Mechanical Systems and 
Signal Processing, 1995 Vol. 9, No. 3, pp. 243-256. 

2. Friswell, M.I., Lees, A.W. and Smart, M.G., Model Updating Tech¬ 
niques Applied to Turbo-Generators Mounted on Flexible Founda¬ 
tions. NAFEMS Second International Conference: Structural Dynam¬ 
ics Modelling Test, Analysis and Correlation, Glasgow: NAFEMS, 
1996 pp. 461-472. 


1041 


3. Lees, A.W. and Friswell, M.L, Estimation of Forces Exerted on Ma¬ 
chine Foundations. Identification in Engineering Systems, Wiltshire; 
The Cromwell Press Ltd., 1996, pp. 793-803. 

4. Smart, M.G,, Friswell, M.L, Lees, A.W. and Prells, U., Errors in 
estimating turbo-generator foundation parameters. In Proceedings 
ISMA21 - Noise and Vibration Engineering, ed. P. Sas, Katholieke 
Universiteit Leuven, Belgium, 1996, Vol. II, pp. 1225-1235 

5. Zanetta, G.A., Identification Methods in the Dynamics of Turbogener¬ 
ator Rotors. The International Conference on Vibrations in Rotating 
Machinery, IMechE, C432/092, 1992, pp. 173-181. 

6. Friswell, M.L and Mottershead, J.E., Finite Element Model Updating 
in Structural Dynamics. Dordrecht, Boston, London: Kluwer Aca¬ 
demic Publishers, 1995. 

7. Natke, H.G., Einfuhrung in die Theorie und Praxis der Zeitreihen- 
und Modalanalyse - Identifikation schwingungsfdhiger elastomechanis- 
cher Systeme. Braunschweig, Wiesbaden: Friedrich Vieweg Sz Sohn, 
1993. 

8. Natke, H.G., Lallement, G., Cottin, N. and Prells, U., Properties of 
Various Residuals within Updating of Mathematical Models. Inverse 
Problems in Engineering, Vol. 1, 1995, pp. 329-348. 

9. Natke, H.G., What is a true mathematical model? - A discussion of 
system and model definitions. Inverse Problems in Engineering, 1995, 
Vol. 1, pp. 267-272. 

10. Gawronski, W. and Natke, H.G., On ARMA Models for Vibrating 
Systems. Probabilistic Engineering Mechanics, 1986, Vol. 1, No. 3, 
pp. 150-156. 

11. Baumeister,J., Stable Solution of Inverse Problems. Braunschweig, 
Wiesbaden: Friedrich Vieweg &; Sohn, 1987. 

ACKNOWLEDGEMENTS 

The authors are indebted to Nuclear Electric Ltd and Magnox Pic for fund¬ 
ing the research project BB/G/40068/A to develop methods which enables 
the estimation of the influence of the foundation on the dynamics of the 
rotor. Dr. Friswell gratefully acknowledges the support of the Engineering 
and Physical Sciences Research Council through the award of an Advanced 
Fellowship. 


1042 


MATHEMATICAL HYSTERESIS MODELS AND 
THEIR APPLICATION TO NONLINEAR 
ISOLATION SYSTEMS 


Y.Q. Ni, J.M. Ko and C.W. Wong 

Department of Civil and Structural Engineering 
The Hong Kong Polytechnic University, Hong, Kong 

Abstract 

Two mathematical hysteresis models, the Duhem-Madelung (DM) 
model and the Preisach model, are introduced to represent the 
hysteretic behavior inherent in nonlinear damping devices. The DM 
model generates the hysteresis with local memory. Making use of the 
Duhem operator, the constitutive relation can be described by single¬ 
valued functions with two variables in transformed state variable 
spaces. This makes it feasible to apply the force-surface nonparametric 
identification technique to hysteretic systems. The Preisach model can 
represent the hysteresis with nonlocal memory. It is particularly 
suitable for describing the selective-memory hysteresis which appears 
in some friction-type isolators. An accurate frequency-domain method 
is developed for analyzing the periodic forced vibration of hysteretic 
isolation systems defined by these models. A case study of wire-cable 
vibration isolator is illustrated. 


1. Introduction 

The dynamic response of a structure is highly dependent on 
the ability of its members and connections to dissipate energy by 
means of hysteretic behavior. The assessment of this behavior 
can be done by means of experimental tests and the use of 
analytical models that take into account the main characteristics 
of this nonlinear mechanism. Although a variety of hysteresis 
models have been proposed in the past decades, many structural 
systems exhibit more complicated hysteretic performance (mainly 
due to stiffness or/and strength degrading) which the models in 
existence are reluctant and even inapplicable to depict [1,2]. On 
the other hand, nonlinear vibration isolation has recently been 
recognized as one of effective vibration control techniques. In 


1043 


particular, hysteretic isolation devices have got wide applications 
owing to their good diy friction damping performance. These 
hysteretic isolators may exhibit very complicated features such 
as asymmetric hysteresis, soft-hardening hysteresis, nonlocal 
selective-memory hysteresis [3-5]. None of the models available 
currently in structural and mechanical areas can represent all 
these hysteresis characteristics. Other more elaborate hysteresis 
models need to be established for this purpose. 

In reality, hysteresis phenomenon occurs in many different 
areas of science, and has been attracting the attention of many 
investigators for a long time. However, the true meaning of 
hysteresis varies from one area to another due to lack of a 
stringent mathematical definition of hysteresis. Fortunately, 
because of the applicative interest and obvious importance of 
hysteresis phenomenon, Russian mathematicians in 1970’s and 
the Western mathematicians in 1980’s, began to study hysteresis 
systematically as a new field of mathematical research [6,7]. They 
also deal with the hysteresis models proposed by physicists and 
engineers in various areas, but they separate these models from 
their physical meanings and formulate them in a purely 
mathematical form by introducing the concept of hysteresis 
operators. Such mathematical exposition and treatment can 
generalize a specific model from a particular area as a general 
mathematical model which is applicable to the description of 
hysteresis in other areas. In this paper, two mathematical 
hysteresis models are introduced and the related problems such 
as identification and response analysis encountered in their 
application to nonlinear isolation systems are addressed. 

2. Definition of Hysteresis 

Hysteresis loops give the most direct indication of hysteresis 
phenomena. But it is intended here to introduce a mathematical 
definition of hysteresis. Let us consider a constitutive law: u r, 
which relates an input variable u(t) and an output variable r\t). 
For a structural or mechanical system, u(t) denotes displacement 
(strain); r{t) represents restoring force (stress); t is time. We can 
define hysteresis as a special type of memoiy-based relation 


1044 


between u(t) and r(t). It appears when the output r[t) is not 
uniquely determined by the input u(t) at the same instant t, but 
instead r(t) depends on the evolution of u in the interval [0, t] and 
possibly also on the initial value ro, i.e. 

r(t) = iR[u(-),ro](t) (1) 

where the memory-based functional iR[u(-),ro](t) is referred to as 
hysteresis operator. In order to exclude viscosity-type memory 
such as those represented by time convolution, we require that 91 
is rate-independent, i.e. that r(t) depends just on the range of u in 
[0, t] and on the order in which values have been attained, not on 
its velocity. In reality, memory effects may be not purely rate 
independent as hysteresis is coupled with viscosity-type effects. 
However, as shown later, in most cases the rate independent 
feature of hysteresis is consistent with experimental findings, 
especially when evolution (variation in time) is not too fast. 

3. Duhem-Madelung (DM) Model 

3.1 Formulation 

The DM model can be defined with or without referring to a 
confined hysteresis region. For the structural or mechanical 
hysteretic systems, it is not necessary to introduce the notion of 
bounded curves because there exists neither the saturation state 
nor the major loop. In this instance, the DM model establishes a 
mapping (named Duhem operator) 91: (u, ro) rby postulating the 


following Cauchy problem [7,8] 

= g, (li.'■) ■ w, (t) - 02(u,r) • u_(t) (2a) 

r(0) = r„ (2b) 

where an overdot denotes the derivative with respect to t; gi{u,r] 
and g 2 {ii,r) are referred to as ascending and descending functions 
(curves) respectively; and 

tijt) = max[0,u(t)] = t [|u(t)| + u(t)] (3a) 

u.(t) = min[0,u(t)] = ^[|u(t)| - u(f)] (3b) 

Eq. (2a) can be rewritten as 

f(f) = p[u,r,sgn(u)]-u(t) (4) 


1045 


u> 0 
u<0 


(5) 


in which the describing function has the form 


g{u,r,sgn{u)] 


fg{u,r,l) = g,{u,r) 
[g[u,r-l) = g^(u,r) 


It is obvious that the DM model is rate independent. In 
addition, it is specially noted that in this constitutive law the 
output r{t) is not directly dependent on the entire history of u{t) 
through [0,t]; but instead depends only on the local histoiy 
covered since the last change of sgn(u) and on the value of the 
output at this switching instant. It means that the output can 
only change its character when the input changes direction. As a 
consequence, the DM model usually represents the hysteresis 
with local memory except that the functions gi(u,?) and g2{u,7) are 
re-specified as hysteresis operators. 

Within the framework of DM formulation, the ascending and 
descending functions gi(u,T] and g 2 (u,r) are just required to fulfill 
suitable regularity conditions and need not to be specified in 
specific expressions, so both the form and parameters of the 
functions can be fine-tuned to match experimental findings. On 
the other hand, the DM formulation can deduce a wide kind of 
differential-type hysteresis models such as Bouc-Wen model, 
Ozdemir’s model, Yar-Hammond bilinear model and Dahfs 


frictional model. For the Bouc-Wen model 

r(t) = K.u{t) + z(t) (6a) 

z{t) = au(t] - P|ti(f)|z(f)|z(t)|'”‘ - YU(t)|z(f)|" (6b) 

it corresponds to the DM model with the specific ascending and 
descending functions as 

grj(u,r) = a + K “ [y + p sgn(r -Ku)]|r -Kup (7a) 

g 2 (w,r) = a + K - [y - p sgn(r -Kw)]|r -ku|” (7b) 

and for the Yar-Hammond bilinear model 

f{t) = {a -y sgn(ii) sgn[r - p sgn(ii)]}ii (8) 

its describing function is independent of u(t) as follows 

5f[u,r,sgn(u)l = gf[r,sgn(u)l = a -y sgn(u)sgn[r - p sgn(u)] (9) 


Hence, the Duhem operator also provides an accessible way 
to construct novel hysteresis models by prescribing specific 


1046 



expressions of the ascending and descending functions. 
Following this approach, it is possible to formulate some models 
which allow the description of special hysteretic characteristics 
observed in experiments, such as soft-hardening hysteresis, 
hardening hysteresis with overlapping loading envelope, and 
asymmetric hysteresis [5]. 

3.2 Identification 

System identification techniques are classified as parametric 
and nonparametric procedures. The parametric identification 
requires that the structure of system model is a priori known. The 
advantage of nonparametric identification methods is that they 
do not require a priori the knowledge of system model. The most 
used nonparametric procedure for nonlinear systems is the force 
mapping (or called force surface) method [9]. This method is 
based on the use of polynomial approximation of nonlinear 
restoring force in terms of two variables—^the displacement u(t) 
and the velocity u{t ). For nonlinear hysteretic systems, however, 
the hysteretic restoring force appears as a multivalued function 
with respect to the variables u{t) and u(t) due to its history- 
dependent and non-holonomic nature. This renders the force 
mapping method inapplicable to hysteretic systems, although 
some efforts have been made to reduce the multivaluedness of 
the force surface [10,11]. 

One of the appealing virtues of the DM model is that it can 
circumvent this difficulty. Making use of the Duhem operator, 
the hysteretic constitutive relation of Eq.(l) is described by two 
continuous, single-valued functions gi(u,r) and g 2 {u,r) in terms of 
the displacement u(t) and the restoring force r{t). Thus, single¬ 
valued “force” surfaces gi{u,r) and g2{u,7) can be formulated in 
the subspaces of the state variables {u,r,gi) and (u,r,g 2 ), and can 
be identified by using the force mapping technique. Following 
this formulation, a nonparametric identification method is 
developed by the authors [12]. In this method, the functions 
g\(u, 7 ) and g2{u,7) are expressed in terms of shifted generalized 
orthogonal polynomials with respect to u and r as follows 

gM,r) = i = 0’'(u)G<''0(r) (10a) 

i=lj=l 


1047 


(10b) 


g^r] = z i(|>,.(r)gf(i>,(u) = 0^(u)G'^'<l>(r) 

I=lj=l 

where = [gf ]^xn =[5'zf]mxn ^re called the expansion- 

coefficient matrices of gi{u,r) and g 2 {u,T]. Some algorithms have 
been proposed to estimate the values of these coefficient matrices 
based on experimentally observed input and output data. It 
should be noted that here the vectors 0(u) and <^{r) are shifted 
generalized orthogonal polynomials [13]. They are formulated on 
the basis of common recurrence relations and orthogonal rule, 
and cover all kinds of individual orthogonal polynomials as well 
as non-orthogonal Taylor series. Consequently, they can obtain 
specific polynomial-approximation solutions of the same.problem 
in terms of Chebyshev, Legendre, Laguerre, Jacobi, Hermite and 
Ultraspherical polynomials and Taylor series as special cases. 

4. Preisach Model 


4.1 Formulation 

The intent of introducing the Preisach model is to supply the 
lack of a suitable hysteresis model in structural and mechanical 
areas, which is both capable of representing nonlocal hysteresis 
and mathematically tractable. Experiments revealed that the 
hysteretic restoring force of some cable-type vibration isolators 
relates mainly to the peak displacements incurred by them in the 
past deformation [3]. It will be shown that the Preisach model is 
especially effective in representing such nonlocal but selective- 
memory hysteresis, in which only some past input extrema (not 
the entire input variations) leave their marks upon future states 
of hysteresis nonlinearities. 

The Preisach model is constructed as a superposition of a 
continuous family of elementary rectangular loops, called relay 
hysteresis operators as shown in Fig. 1. That is [7,14], 

r(f) ='R[u(-)](f) = j||^(a,P)y„D[u(t)]dadf5 (11) 

a>P 

where ia(a,|3) > 0 is a weight function, usually with support on a 
bounded set in the (a,p)-plane, named Preisach plane; Ya.p[^(^)] is 
the relay hysteresis operator with thresholds a > p. Outputs of 


1048 



these simplest hysteresis operators have only two values +1 and 
-1, so can be interpreted as two-position relays with “up” and 
“down” positions corresponding to ya,p[^^(^)l=+l and 




1 

+1 

p 



a 






-1 



Fig. 1 Relay Hysteresis Operator 


Hence, the Preisach model of Eq.(ll) can be interpreted as a 
spectral decomposition of the complicated hysteresis operator iR, 
that usually has nonlocal memory, into the simplest hysteresis 
operators 7ct,p with local memory. In the following, we illustrate 
how the model depicts the nonlocal selective-memory feature. 

Consider a triangle T in the half-plane a > (3 as shown in 
Fig.2. It is assumed that the weight function )i(a,p) is confined in 
the triangle T, i.e. }i(a,p) is equal to zero outside T. Following the 
Preisach formulation, at any time instant t, the triangle T can be 
subdivided into two sets: S^(t) consisting of points (a,P) for which 
the corresponding Yc^.p-operators are in the “up” position; and S^(t) 
consisting of points {a,P) for which the corresponding Ya.p“ 
operators are in the “down” position. The interface L(t) between 
S^(t) and Sr(t) is a staircase line whose vertices have a and p 
coordinates coinciding respectively with local maxima and 
minima of input at previous instants of time. The nonlocal 
selective-memory is stored in this way. Thus, the output r(^) a-t 
any instant t can be expressed equivalently as [14] 

r{t)= J|^(a,P)dadp- |jM,(a,P)dadp (12) 

S'(£) S'(t) 

It should be noted that the Preisach model does not 
accumulate all past extremum values of input. Some of them can 
be wiped out by sequent input variations following the wiping-out 
property {deletion rule): each local input maximum wipes out the 


1049 


vertices of L(t) whose a-coordinates are below this maximum, and 
each local minimum wipes out the vertices whose p-coordinates 
are above this minimum. In other words, only the alternating 
series of dominant input extrema are stored by the Preisach 
model; all other input extrema are erased. 



Fig. 2 Input Sequence and Preisach Plane 


4.2 Identification 


It is seen from Eq.(l 1) that the Preisach model is governed by 
the weight function |i(a,p) after determining L(t) which depends 
on the input sequence. !i(a,P) is a single-valued function with 
respect to two variables a and p. Hence, the aforementioned 
nonparametric identification method can be also implemented to 
identify iLL(a,p) by expanding it in a similar expression to Eq.(lO). 
An alternating approach is to define the following function 


H'(a',P') = |j|a(a,p)dadp = ^.[^ ^(ot,P)da]dp (13) 


r(cc'.3') 


where T(a',p') is the triangle formed by the intersection of the 
line a = a', p = p' and a = p. Differentiating Eq.(13) yields 


^(a',P') = -- 


aa'ap' 


(14) 


Thus, the force mapping identification technique can be applied 
to determine H(a,p) consistent with the experimental data, and 
then ^(a,p) is obtained by Eq.(14). 


5. Steady-State Response Analysis 

Hysteretic systems are strongly nonlinear. A study of the 
steady-state oscillation is one of the classical problems of 


1050 



nonlinear systems. Usually, the dynamic behavior of a nonlinear 
system is represented by its resonant frequency and frequency 
response characteristics. In the following, an accurate frequency- 
domain method accommodating multiple harmonics is developed 
to analyze the periodically forced response of hysteretic systems 
defined by mathematical hysteresis models. 

Fig.3 shows a single-degree-of-freedom hysteretic oscillator 
with mass m, viscous damping coefficient c, and linear stiffness 
ky subjected to an external excitation F(t), for which the governing 
equation of motion is 

m • u(t) + c • u(t) + k ' u[t) + r{t) = F(t) (15) 

where the hysteretic restoring force r(t) is represented by the DM 
model as Eq.(4). It is worth noting that for the kinetic equation 
Eq.(15), the excitation is F(t) and the response is u(t); and for the 
hysteretic constitutive law Eq.(4), u(t) is input and r{t) is output. 
The causal relationship is different. 



Fig. 3 Single-Degree-of-Freedom Hysteretic System 

Due to the hereditary nature of the hysteresis model, it is 
difficult to directly solve the kinetic equation Eq.(15) by iteration. 
Here, Eq.(15) is only used to establish the relation between the 
harmonic components of u(t) and r(t). Suppose that the system is 
subjected to a general periodic excitation F[t) with known 
harmonic components F={Fq F^ F 2 ••• F 2 . The multi¬ 
harmonic steady-state response can be expressed as 

a ^ ^ . 

u{t) = ~+ aj cos j(Dt+ aj sin jcot (16) 

2 j=i 


1051 


in which a={ao a 2 ••• a^v ^2 unknown 

vector containing the harmonic components of u(t). Introducing 

Eq.(16) into Eq.(15) and using the Galerkin method provide 

ro^F^-k-a^ (17a) 

Tj = Fj -c-citj-a] [j = 1,2, ••• , N) (17b) 

Tj =Fj +c-coj-a^. -(/c-m-coV^)-a* (j = 1, 2, ,N) (17c) 

where r={ro q ••• r^}'^ is the harmonic vector of 

the hysteretic restoring force r(t). Referring to the hysteretic 
constitutive law, we define the determining equation as 

D(t) = r{t) - g[u,r,sgn(u)] • u{t) (18) 

When a is the solution of u(t), applying the Galerkin method 
into Eq.(18) and considering Eq.(17) achieve 

d{a) = 0 (19) 

where the vector d{a)={dQ ^3 --dj^ d^ d^ "• is comprised 

of the harmonic components of D(t) corresponding to a. An 

efficient procedure to seek the solution of Eq.(19) is the 
Levenberg-Marquardt algorithm with the iteration formula 

where the Jacobian matrix J[a(^)] = dd(a)/da\a=a^^) ; 9 ^ is the 

Levenberg-Marquardt parameter and I is identity matrix. 

At each iteration, the function vector and Jacobian 

matrix should be recalculated with updated values of 

Here, a frequency/time domain alternation scheme by FFT is 
introduced to evaluate the values of d(a) and J[a) at d{a) 

and dd[a)/da are known to be the Fourier expansion coefficients 
of D(t) and dD(t)/ da respectively. For a given a(^) and known F, the 
corresponding r[a(^)] is obtained from Eq.(17), and the inverse 
FFT is implemented for and r[a(^)] to obtain all the time 
domain discrete values of u{t], u(t) , r(t) and f(t) over an integral 
period. Then the time domain discrete values of the function 
D{u,u,r,r,t), corresponding to a=a(^), are evaluated from Eq.(18). 
Making use of forward FFT to these time domain discrete values 
of D{u,u,r,r,t) , the values of function vector d[a(^)] are obtained. 


1052 


Similarly, the partial differential dD{t) / da can be analytically 
evaluated in the time domain. Forward FFT to the time domain 
values of dD(t) / da at ct=a(^) gives rise to dd[ai^)]/da. 

6. Case Study 

Wire-cable vibration isolators are typical hysteretic damping 
devices. Dynamic tests show that their hysteresis behaviors are 
almost independent of the frequency in the tested frequency 
range [4,15]. Experimental study and parametric modelling of a 
wire-cable isolator have been carried out [5]. Fig.4 shows the 
experimental hysteresis loops in shear mode. It is seen that for 
relatively small deformations, the isolator exhibits softening 
hysteresis loops. When large displacements are imposed, the 
stiffness of the loops becomes smoothly hard. This nature is 
referred to as soft-hardening hysteresis. Based on the Bouc-Wen 
model, a parametric identification was performed to model these 
hysteresis loops, but the result is unsatisfactory. This is due to 
the fact that the Bouc-Wen model cannot represent such soft- 
hardening nature of hysteresis. 



Fig. 4 Experimental Hysteresis Loops 


We now use the DM model to represent these hysteresis 
loops, and perform a nonparametric identification to determine 
the functions gi(u,7) and g2(u,7). The simplest Taylor series are 
adopted, i.e. (i),(r) = (r / and (|)j(u) = (u j UqY~^ (ro = 20.0 and 
uo = 2.0). Fig.5 shows the identified “force” surfaces of gi{u,T) and 
g 2 (u,r) by taking m=n~S. Fig.6 presents the theoretical hysteresis 


1053 




loops generated by the DM model using the identified g\(u,T) and 
g 2 (u,}). It is seen that the modeled hysteresis loops are agreeable 
to the observed loops. In particular, the soft-hardening nature is 
reflected in the modeled hysteresis loops. 


-200 





Fig. 6 Modeled Hysteresis Loops by DM Model 




After performing the modelling of hysteretic behavior, the 
dynamic responses of hysteretic systems can be predicted by the 
developed method. Fig.7 shows a vibration isolation system 
installed with wire-cable isolators in shear mode. It is subjected 
to harmonic ground acceleration excitation Xg[t)=Acos2Kft. The 

equation of motion of the system is expressed as 

m-u(t) +K ■r(t) =-rri'Xg(t) (21) 

where m is the mass of the system; K is number of the isolators 
installed. u{t) is the displacement of the system relative to the 
ground. r(t) is the restoring force of each isolator and has been 
determined from nonparametric identification. 

Fig.8 illustrates the predicted frequency-response curves of 
the relative displacement when 7n=6kg and K=2. The excitation 
amplitude A is taken as 0.25g, O.SOg, 0.35g, 0.40g and 0.45g 
respectively. The frequency-response curves show clearly the 
nonlinear nature of the wire-cable isolation system. 



i 

I 


Xg(t) 


Fig. 7 Vibration Isolation System with Wire-Cable Isolators 



Fig. 8 Frequency Response Curves of Relative Displacement 


1055 




7. Concluding Remarks 


This paper reports a preliminaiy work of introducing the 
mathematical hysteresis models in structural and mechanical 
areas. It is shown that a wide kind of differential hysteresis 
models, which are extensively used at present, can be derived 
from the Duhem-Madelung (DM) model. Thus, the mathematical 
properties concerning the DM model are also possessed by these 
models. Two potential advantages appear when the DM 
formulation is used. Firstly, it allows to apply the force mapping 
technique to hysteretic systems. Secondly, it provides an 
approach to construct novel differential models which reflect 
some special hysteretic characteristics. The Preisach model is 
shown to be capable of representing nonlocal hysteresis and 
mathematically tractable. It offers a more accurate description of 
several observed hysteretic phenomena. Emphasis is placed on 
demonstrating the selective-memory nature of this hysteresis 
model. The case study based on experimental data of a wire- 
cable isolator has shown the applicability of the mathematical 
hysteresis model, and the validity of the steady-state response 
analysis method proposed in the present paper. 

Acknowledgment: This study was supported in part by the 
Hong Kong Research Grants Council (RGC) and partly by The 
Hong Kong Polytechnic University. These supports are gratefully 
acknowledged. 


References 

1. Azevedo, J. and Calado, L., ‘‘Hysteretic behaviour of steel 
members: analytical models and experimental tests”, J. 
Construct Steel Research, 1994, 29, 71-94. 

2. Kayvani, K. and Barzegar, F., “Hysteretic modelling of tubular 
members and offshore platforms”, Eng. Struct, 1996, 18, 93- 
101 . 

3. Lo, H.R., Hammond, J.K. and Sainsbury, M.G., “Nonlinear 
system identification and modelling with application to an 
isolator with hysteresis”, Proc. 6th Int modal Anal Conf., 
Kissimmee, Florida, 1988, Vol.II, 1453-1459. 


1056 


4. Demetriades, G.F., Constantinou, M.C. and Reinhorn, A.M., 
“Study of wire rope systems for seismic protection of 
equipment in buildings”, Eng. Struct, 1993, 15, 321-334. 

5. Ni, Y.Q., “Dynamic response and system identification of 
nonlinear hysteretic systems”, PhD Dissertation, The Hong 
Kong Polytechnic University, Hong, Kong, November 1996. 

6. Krasnosefskii, M.A. and Pokrovskii, A.V., Systems with 
Hysteresis, translated from the Russian by M. Niezgodka, 
Springer-Verlag, Berlin, 1989. 

7. Visintin, A., Differential Models of Hysteresis, Springer-Verlag, 
Berlin, 1994, 

8. Macki, J.W., Nistri, P. and Zecca, P., “Mathematical models 
for hysteresis”, SIAM Review, 1993, 35, 94-123. 

9. Masri, S.F. and Caughey, T.K., “A nonparametric 

identification technique for nonlinear dynamic problems”, J. 
Appl Mech, ASME, 1979, 46, 433-447. 

10. Lo, H.R. and Hammond, J.K., “Nonlinear system identification 
using the surface of nonlinearity form: discussion on 
parameter estimation and some related problems”, Proc. 3rd 
Int Conf Recent Adv. Struct. Dyn., Southampton, UK, 1988, 
339-348. 

11. Benedettini, F., Capecchi, D. and Vestroni, F., “Identification 
of hysteretic oscillators under earthquake loading by 
nonparametric models”, J. Eng. Mech., ASCE, 1995, 121, 606- 
612. 

12. Ni, Y.Q., Ko, J.M. and Wong, C.W., “Modelling and 
identification of nonlinear hysteretic vibration isolators”. 
Accepted to SPJE’s 4th Annual Symposium on Smart Structures 
and Materials : Passive Damping and Isolation, 3-6 March 
1997, San Diego, USA. 

13. Ni, Y.Q., Wong, C.W. and Ko, J.M., “The generalized 
orthogonal polynomial (GOP) method for the stability analysis 
of periodic systems”, Proc. Int. Conf. Comput. Methods Struct. 
Geotech. Eng., Hong Kong, 1994, Vol.II, 464-469. 

14. Mayergoyz, I.D., Mathematical Models of Hysteresis, Springer- 
Verlag, New York, 1991. 

15. Ko, J.M., Ni, Y.Q. and Tian, Q.L., “Hysteretic behavior and 
empirical modeling of a wire-cable vibration isolator”, Int. J. 
Anal. Exp. Modal Anal, 1992, 7, 111-127. 


1057 



1058 


The identification of turbogenerator foundation models 
from run-down data 


M Smart, M I Friswell, A W Lees, U Prells 

Department of Mechanical Engineering 
University of Wales Swansea, Swansea SA2 8PP UK 
email: m.smart@swansea.ac.nk 


ABSTRACT 

The trend of placing turbines in modern power stations on flexi¬ 
ble steel foundations means that the foundations exert a considerable 
influence on the dynamics of the system. In general, the complex¬ 
ity of the foundations means that models are not available a priori, 
but rather need to be identified. One way of doing this is to use 
the measured responses of the foundation at the bearings to the 
synchronous excitation obtained when the rotor is run down. This 
paper discusses the implementation of such an estimation technique, 
based on an accurate model of the rotor and state of unbalance, and 
some knowledge of the dynamics of the bearings. The effect of errors 
in the bearing model and response measurements on the identified 
parameters is considered, and the instrumental variable method is 
suggested as one means of correcting them. 


1 INTRODUCTION 

The cost of failure of a typical turbine in a modern power station is 
very high, and therefore development of condition monitoring techniques 
for such machines is an active area of research. Condition monitoring relies 
on measuring machine vibrations and using them to locate and quantify 
faults, which obviously requires an accurate dynamic model of the ma¬ 
chine. Although the dynamic characteristics of rotors are generally well 
understood, the foundations on which they rest are not. Since the founda¬ 
tions are often quite flexible, they can contribute considerably to the rotor’s 
dynamic behaviour. 

Finite element modelling has been attempted but the complexity of 
the foundations, and the fact that they often differ substantially from the 
original drawings rendered the technique generally unsuccessful[1]. Exper¬ 
imental modal analysis is another possible solution, but this requires that 
the rotor be removed from the foundation, and that all casings remain in 
place, which is not practical for existing power plant. However, mainte¬ 
nance procedures require that rotors are run down at regular intervals and 
this procedure provides forcing to the foundation over the frequency range 


1059 


of operation. By measuring the response at the bearing pedestals (which 
is already performed for condition monitoring purposes) an input-output 
relation for the foundation may be obtained. 

Lees [2] developed a least-squares method to calculate the foundation 
parameters by assuming that an accurate model exists for the rotor, that 
the state of unbalance is known from balancing runs, and that the dynamic 
stiffness matrices of the bearings can be calculated. Although bearing mod¬ 
els are not in fact well characterised, Lees and Friswell[3] showed that the 
parameter estimates are only sensitive to the bearing stiffnesses over limited 
frequency ranges, which can be calculated. 

Feng and Hahn[4] followed a similar approach but added extra informa¬ 
tion by measuring the displacements of the shaft. Zanetta[5] also measured 
the shaft displacements but included the bearing characteristics as param¬ 
eters to be estimated. Although any extra information is desirable in a 
parameter estimation routine the equipment necessary to measure these 
quantities only exists in the newer power stations, and it was desired to 
make the method applicable to older plant as well. In the analysis presented 
here, the measured data consists of the motion of the bearing pedestals in 
the horizontal and vertical directions, although not necessarily in both di¬ 
rections at every bearing. 

2 THEORY 

2.1 Force estimation 

If D is the dynamic stiffness matrix of a structure defined as 

D{u) K ^luC -oj^M (1) 

where M,C,K are the mass, damping and stiffness matrices then 

Dx = f (2) 

where x is response and f force. Referring to figure 1 it is seen that the rotor 
is connected to the foundation via the bearings. It is assumed that good 
models exist for both rotor and bearings, and that the state of unbalance 
is known. The implications of these assumptions will be discussed later. 

The dynamic stiffness equation for the whole system may be written as 



The subscripts r and / refer to the rotor and foundation degrees-of-freedom 
respectively, u refers to the unbalance forces and b to the bearing forces. 


1060 


ROTOR 



Figure 1: Rotor-bearing system 


There is a negative sign before the bearing forces ft, since they refer to the 
forces acting on the bearings. The foundation d.o.f are those where the re¬ 
sponses are measured, in other words no internal d.o.f are represented. Df 
therefore represents a reduced dynamic stiffness matrix. The response mea¬ 
surements will not be the total vibration level at the bearings but rather the 
vibrations at once-per-revolution and it is assumed that no dynamic forces 
at this frequency will be transmitted to the foundation via the substructure 
onto which it is fixed. 

The equation for the bearings in the global coordinate system is 


/ Dbrr Dbrf W ^7- ^ ^ fbT\ 
V Dij, Dill )\^s) \fbs) 


(4) 


This assumes that the bearings behave as complex springs, in other 
words they have negligible inertia and no internal d.o.f. Substituting (4) 
into (3) we have 


/ Dt„ Dtrl \ ( ^'■ \ = ( 

V A/r A// J [ J A ° / 

where 

/ Dtrl 0 \ f Dirr Di^l \\ 

[Dtlr DtllJ-[[ 0 DlJ + [Diir Dill)) 


(5) 

( 6 ) 


1061 


and where subscript t refers to the total model. Solving equation (5) for 
Xr leads to 

Xr — J^trr ifur ~~ 

and solving equation (4) for ft / yields 

fbf — ^fhr ~ ~^brr^r ~ ^brf^f (S) 

All quantities in equation (8) are known either from assumed models {Dr, D^) 
or experiment {xf, /^r)- This calculated force fbf may then be used to¬ 
gether with the measured responses to estimate the foundation parameters. 

2.2 Foundation parameter estimation 

Once the forces have been estimated, the foundation parameters must 
be derived. The dynamic stiffness equation for the foundation is 


DfXf — fbf (9) 

Although D/ is a reduced stiffness matrix it is assumed that it has the 
form of equation (1). Therefore equation (9) may be written as 

W{u)v = fbf (10) 

where is a column vector formed from the elements of M, C and K and 
W is a matrix formed from the response vector which depends explicitly on 
u. Clearly this is an under-determined set of equations, but by taking mea¬ 
surements at many frequencies it may be made over-determined, and thus 
solvable in a least squares sense. Since the magnitude of the mass, damp¬ 
ing and stiffness elements normally differ by several orders-of-magnitude, it 
was found expedient to scale the mass parameters by o;^, and the damping 
parameters by uJ, where u is the mean value of the frequency. 

2.3 Errors in estimates 

It is necessary now to examine the effect of errors on the parameter esti¬ 
mates. Equation (10) is of the form Ax = b, where A has dimension mxn. 
In this particular case, A depends on the measured response Xf, whilst b 
depends on the measured response, applied unbalance, and assumed rotor 
and bearing models. Therefore the estimated parameters will be sensitive 
to the following errors: 

1. Errors in the rotor model 

2. Errors in the bearing model 

3. Errors in the state of unbalance 


1062 


4. Errors in the measured foundation response 

The rotor model is generally well known, as is the state of unbalance, 
so the main source of error in the estimates is due to measurement noise 
and bearing uncertainty. If the least squares problem is formulated as 


Ax = b (11) 

then 

(Aq + Ajv)ic = bo + b/i/ (12) 

where the subscript N refers to noise and 0 to data which is noise-free. The 
least-squares estimate is given by 

X = (Aq Ao + A^Aiv + + A^Ao) ^(Aq + Ayv)^(bo + b;v) (13) 

Even if the noise on the outputs is uncorrelated with the noise on the inputs 
the expected value of x does not equal that of its estimate: 

E[x] - E[x] 0 (14) 

In other words the estimate x is biased [6]. In order to reduce the bias of 
the estimates, the instrumental variables method can be used. Essentially, 
it requires the use of a matrix that is uncorrelated with the noise on the 
outputs, but which is strongly correlated with the noise-free measurements 
themselves. If W is the instrumental variable matrix, then 

W'^Ax = W'^b (15) 

Expanding 

i = (W^Ao + + b^) (16) 

This means that E[x] = x, in other words unbiased estimates result. 
Fritzen[7] suggested an iterative method for solving for the parameters. 
Initially, equation (10) is solved in a least-squares sense, and the values 
of the estimated parameters are used to calculate outputs for the model. 
These outputs are then used to create W in the same way as the original 
outputs were used to create A, new estimates are obtained, and if neces¬ 
sary the process is repeated. Experience seems to suggest good convergence 
properties[7]. 


3 SIMULATION 

The method under discussion in this paper was tested on a model of a 
small test rig located at Aston University, Birmingham. This consists of 
a steel shaft approximately 1.1m long, with nominal diameter 38mm. The 
shaft is supported at either end by a journal bearing of diameter 100mm, 


1063 


1 2 3 4 5 6 7 8 9 10 11 12 



Figure 2: Rotor-bearing system 

L/D ratio of 0.3 and clearance of 25/.tm. There are two shrink-fitted bal¬ 
ancing discs for balancing runs. Each bearing is supported on a flexible 
pedestal to simulate the flexible foundations encountered in power station 
turbines. At present these pedestals are bolted onto a massive lathe bed. 
The rotor is powered by a DC motor attached via a belt to a driving pulley, 
which is in turn attached via a flexible coupling to the main rotor shaft. 

A schematic of the rig is shown in figure 2. Dimensions of each station 
and material properties are given in table 1. A finite element model of the 
rotor with 23 elements was created and short bearing theory was used to 
obtain values for the bearing stiffness and damping[8]. 

The pedestals themselves consist of two rectangular steel plates, 600mm 
X 150mm which have two channels cut into them, and which are supported 
on knife-edges (figure 3). The vertical stiffness arises from the hinge effect 
of the channels, whilst the horizontal stiffness is as a result of the shaft 
centre tilting under an applied load. Treating the supports as beams, the 
theoretical stiffnesses are: 

Ky = 0.^5MN/m !<:, = 1.5MiV/m 

where x and y refer to the horizontal and vertical directions respectively. 
The masses and damping factors were taken as: 

il/4 = My = 50/cp Ca: = Cy = 150iV • s/m 

The estimation theory was tested using this model. The finite element 
model was used to generate responses at the bearings for frequencies from 


1064 









Table 1: Table of rotor rig properties 


Shaft Properties 

Station 

Length (mm) 

Diameter (mm) 

E (GPa) 

P (kg/m^) 

1 

6.35 

38.1 

200 

7850 

2 

25.4 

77.57 

200 

7850 

3 

50.8 

38.1 

200 

7850 

4 

203.2 

100 

200 

7850 

5 

177.8 

38.1 

200 

7850 

6 

50.8 

116.8 

200 

7850 

7 

76.2 

38.1 

200 

7850 

8 

76.2 

109.7 

200 

7850 

9 

76.2 

38.1 

200 

7850 

10 

50.8 

102.9 

200 

7850 

11 

177.8 

38.1 

200 

7850 

12 

203.2 

100 

200 

7850 

Balancing discs 

Station 

Length (mm) 

Diameter (mm) Unbalance (kg • m) 


6 

25.4 

203.2 

0.001 



25.4 

203.2 

0.001 



0 to 30 Hz with a spacing of 0.1 Hz. The responses were corrupted by 
normally distributed random noise with zero mean and standard deviation 
of 0.1% of the maximum response amplitude (applied to both real and 
imaginary parts of the response). At each frequency the bearing static 
forces were disturbed by noise drawn from a uniform distribution spanning 
an interval of 20% of the force magnitude, to introduce uncertainty into the 
bearing parameters. The unbalance was assumed to be exactly known. A 
series of 30 runs was performed, foundation parameter estimates calculated 
and the mean and standard deviations of these estimates obtained. 

The magnitudes of the responses at both bearings are given in figure 4, 
which show that there are four critical speeds in the frequency range under 
consideration. A sample of true and estimated forces in the bearings are 
shown in figures 5 and 6. 

The means {fi) and standard deviations (a) of the least-square (LS) 
and instrumental variable (IV) estimates for the foundation parameters 
are shown in table 2. 








Displacement (m) 



Figure 3: Flexible bearing supports 



Figure 4: Magnitudes of responses at bearings 


1066 

















Table 2; Parameter estimates for foundations showing uncertainty 



1069 








4 DISCUSSION 


The results in table 2 show a clear improvement in parameter estimates 
when the instrumental variable method is used. There is a clear bias in 
the least-squares estimates which is significantly less when the instrumen¬ 
tal variable method is employed. Also, despite the fact that the bearing 
parameters are assumed to be seriously in error, the estimates appear to be 
insensitive to them. This will be true provided that the bearings are much 
stiffer than the foundation (a reasonable assumption in practice). It does 
appear however that in some cases the standard deviation of the instru¬ 
mental variable estimate is larger than that of the least-squares estimate, 
a fact which warrants further investigation. 

As far as the rotor model is concerned, impact tests, which are per¬ 
formed on rotors prior to them entering service, normally give experimental 
frequencies which are within a few percent of the theoretical ones. Thus 
the assumption that the rotor model is accurately known would appear to 
be reasonable. 

The state of unbalance may in theory be established from a balancing 
run. If two successive run-downs are performed, one due to the unknown 
system unbalance and one with known balance weights attached, then pro¬ 
vided the system is linear the response measurements may be vectorially 
subtracted to give the response due to the known balance weights alone. 
In order to ascertain the effect of unbalance uncertainty on the parameter 
estimates, one run was performed assuming no error in the unbalance. It 
should be noted that this assumes that the system is linear. 

5 CONCLUSIONS 

A method of estimating turbogenerator foundation parameters from 
potentially noisy measurement data is demonstrated. It is shown that 
making use of the instrumental variable method reduces the bias in the 
estimates and improves them quite significantly. 

6 ACKNOWLEDGEMENT 

The authors wish to acknowledge the support and funding of Nuclear 
Electric Ltd and Magnox Electric Pic. Dr Friswell wishes to acknowledge 
the support of the Engineering and Physical Sciences Research Council 
through the award of an advanced fellowship. 


1070 


REFERENCES 


[1] A. W. Lees and I. C. Simpson. The dynamics of turbo-alternator foun¬ 
dations: Paper C6/83. In Conference on steam and gas turbine founda¬ 
tions and shaft alignment, Bury St Edmunds, 1983, IMechE, pp37-44. 

[2] A. W. Lees. The least squares method applied to identify ro¬ 
tor/foundation parameters: Paper C306/88. In Proceedings of the Inter¬ 
national Conference on Vibrations in Rotating Machinery, Edinburgh, 
1988, IMechE, pp209-216. 

[3] M. 1. Friswell and A. W. Lees. Estimation of forces exerted on machine 
foundations. In M. 1. Friswell and J. E. Mottershead, editors, Interna¬ 
tional Conference on Identification in Engineering Systems, Swansea, 
1996, pp793-803. 

[4] N. S. Feng and E. J. Hahn. Including foundation effects on the vibra¬ 
tion behaviour of rotating machinery. Mechanical Systems and Signal 
Processing, 1995, 9, pp243-256. 

[5] G. A. Zanetta. Identification methods in the dynamics of turbogener¬ 
ator rotors: Paper C432/092. In IMechE Conference on Vibrations in 
rotating machinery, Bath, 1992. IMechE, ppl73-181. 

[6] J. Schoukens and R. Pintelon. Identification of linear systems. Perga- 
mon Press, 1991. 

[7] C. P. Fritzen. Identification of mass, damping and stiffness matrices of 
mechanical systems. Journal of Vibration, Acoustics, Stress and Relia¬ 
bility if Design, 1986 108, pp9-17. 

[8] D. M. Smith. Journal bearings in Turbomachinery. Chapman and Hail, 
1969. 


1071 



1072 


SHELL MODE NOISE IN RECIPROCATING REFRIGERATION 
COMPRESSORS 


Ciineyt Oztiirk and Aydin Bahadir 
Tiirk Elektrik Endiistrisi A.$ 

R&D Department 

Davutpa§a, Litres Yolu, Topkapi -34020, Istanbul, Turkey 


ABSTRACT 

This study describes the successful endeavor to understand the causes of noise 
that appear on the shell modes of the reciprocating refrigeration compressors. 
The compressor shell is generally considered as the acoustic enclosure that 
reflects the acoustic energy back into the compressor cavity but also as the 
transmitter and radiator of the transmitted acoustic energy that could be 
radiated into the air or transmitted to the structure. Vibrations of the 
compressor shell can easily be characterized in terms of the modal parameters 
that consist of the natural frequencies, mode shapes and damping coefficients. 
The noise source harmonics and the shell resonances couple to produce the 
shell noise and vibration. The harmonic spacing is equal to the basic pumping 
frequency. Results of the studies indicate that important natural frequencies of 
the compressor shell usually stay between 2000-6000 Hz interval. The 
important natural frequencies are first natural frequencies in the lower range 
with the longer wavelengths that radiate well. 


INTRODUCTION 

Compressor noise sources are those processes where certain portions are 
separated from the desired energy flow and transmitted through the internal 
components of compressor to the hermetic shell where it is radiated from the 
shell as airborne noise on vibration of supporting structure will eventually 
radiate noise from some portion of the structure. Noise sources of the 
reciprocating refrigeration compressors can be classified as motor noise, 
compression process noise and valve port flow noise. 


1073 


In reciprocating compressors there is very high density of noise harmonics even 
though they decay in amplitude at high frequencies. Generally, these noise 
source harmonics and the shell resonances couple to produce shell noise and 
vibration 


NOISE GENERATION MECHANISM OF THE RECIPROCATING 
REFRIGERATION COMPRESSORS 

Significance of the problem 

The results of the sound radiation characteristics shown at figure 1 indicated 
that certain high amplitude frequency components had very distinctive sound 
radiation patterns. It was suspected these frequencies correspond to excitation 
of either structural resonances of the compressor shell or acoustic resonances of 
the interior cavity space. Resonances those amplify the noise and vibration 
caused by pumping harmonics of a compressor and thus can be the cause of 
significant noise problems. 


Sound power - A-Wcighted 



—I-,-1-,-;-P——r— f —!——'I' "I '"'I —I- ”r ' i i 

50.0 63.0 80,0 100,0 125,0 160,0 200,0 250,0 315,0 400,0 500,0 630,0 800,0 l,0k t,3k 1,6k 2,0k 2,5k 4,0k S,0k 63k S,0k 10,0k 

Frequency [Hz] 


Figure 1, Noise Radiation Characteristics of Reciprocating Refrigeration 
Compressor. 

Noise sources 

Noise in a compressor is generated during cyclic compression, discharge, 
expansion and suction process. The character of noise sources is harmonic due 
to periodic nature of the compression process. These harmonics are present in 
the compression chamber, pressure time history and loading of the compressor 
through drive system. The motor can not provide immediate response to load 
harmonics and load balance is obtained at the expense of acceleration and 


1074 


































deceleration of the motor drive system. Harmonic vibrations of the motor drive 
system can then excite the resonant response in the compressor components 
that can transmit the acoustic energy in very efficient way. The rest of the noise 
sources are, turbulent nature of flow depending to passage through valve ports, 
valve impacts on their seats and possible amplification when matched with 
mechanical resonances. 

MECHANICAL FEATURES OF SHELL 
Mechanical features of the compressor 

The hermetically sealed motor compressor comprises in general a motor 
compressor unit including a motor assembly mounted with a frame and a sealed 
housing within which the compressor is supported by means of plurality of coil 
springs each having one end spring with the frame and the other end connected 
with the interior of the housing. 

The refrigerant gas as it is compressed in the cylinder is discharged through the 
discharge chamber in the cylinder head into the discharge muffler. The 
discharge muffler is generally mounted on the cylinder head attached in 
covering relation to an end face of the cylinder. Where the sealed casing is 
spherical in shape for better noise suppression, an upper end of the cylinder 
head tends to interfere with an inner wall surface of the casing, a disadvantage 
that can only be eliminated by increasing the size of scaled casing for providing 
a desired hermetically sealed motor. 

Compressor Shell 

The shell is easily be characterized with the modal parameters. The ideal shell 
should be designed in a way that keep all the excitation frequencies at the mass 
controlled region of all its modes. But, depending on the very tight constraints 
that come with the gas dynamics and motor locations, it is not allowed to be 
flexible during the design of shell. As a consequence of the existing design 
limitations natural frequencies of the shell usually fall between 1000-5000 Hz. 

SHELL RADIATION 

Figure 1 illustrates how the sound pressure level of a pumping harmonic can 
increase as it nears a resonant frequency. The sound pressure level of the 
pumping harmonic increases around the shell resonances. The resonance 
generally radiates primarily from the large flat sides of the compressor. There 
are three major acoustic cavity resonances 400, 500 and 630 Hz and four major 
structural resonances of the compressor shell: around 1.6 K, 2 K, 2.5 and 3.2 
K. Hz at which noise radiates well in certain directions. 


1075 


SHELL RESONANCES 


Figures 2, 3 and 4 illustrate the frequency responses of the compressor shell 
when measured with the impact hammer method. Figure 2 is the response to 
the excitation in x direction, figure 3 is for the excitation direction and figure 4 
is for the excitation in z direction. 



Figure 2, Frequency response of the compressor shell when excited in X 
direction. 



Figure 3, Frequency response of the compressor shell when excited in Y 
direction. 


1076 




Figure 4, Frequency response of the compressor shell when excited in Z 
direction. 


To verify the hypothesis that resonances were contributing to some of the noise 
problems of the reciprocating piston compressor, a modal analysis of the shell 
and interior cavity was performed. 

For the shell modal analysis, the accelerometer to measure the response 
remained stationary, while the impact location was moved. The test was 
performed in this manner for convenience since it was easier to fix the 
accelerometer in one location and strike the compressor with force hammer at 
each grid point to obtain transfer function for each measurement location. 
Identical results are obtained if impact occurred at a single point and the 
response was measured at each measurement location. Preliminary test were 
performed initially to identify an appropriate measurement location at which all 
important natural frequencies of shell are detectable. Several force input and 
response locations were evaluated to determine the best location to mount 
accelerometer to measure the shell response. 

The shell resonances are also calculated by using the Structural FEM analysis. 
Table-1 lists the natural frequencies predicted in these studies. During the finite 
element analysis, the models of the compressor were built, based on the CAD 
models. The shell consists of 7500 elements. The mesh densities are quite 
adequate for the structural analysis in the frequency range of interest. In order 
to investigate the possible influence of the crank mechanism on the natural 
frequencies of the shell, a simple model of the crank mechanism was introduced 
to the FE model. During the calculations, the crank mechanism was simplified 
as a rigid block with certain mass and rotary inertia and modeled with solid 
elements. The shell and the crank mechanism have been suspended with the 


1077 



springs from 3 positions and in all 6 transitional and rotational directions. 
Depending on the negligible spring effect on the longitudinal direction, 
estimated values have been used in 5 directions. The FE model has been 
assumed to be free-free. 


Mode 

# 

Frequency 

Hz 

Mode 

# 

Frequency 

Hz 

1 

1997 

21 

4716 

2 


22 

4925 

3 

2293 

23 

4994 

4 


24 


5 


25 

5159 

6 


26 

5454 

7 

2889 

27 

5476 

8 

3258 

28 


9 


29 

EESI^HI 

10 


30 

5783 

11 

3376 

31 

5936 

12 

3551 

32 

5999 

13 

3577 

33 

6035 

14 


34 

6055 

15 

3788 

35 

6183 

16 

3958 

36 

6237 

17 

4383 

37 

6314 

18 

4481 

38 


19 

4644 

39 


20 

4702 

40 

6701 1 


Table-1, Calculated Natural Frequencies of the Shell 

Figures 5, 6, 7 and 8 illustrate how the mode shapes of the shell vary at the 
mode frequencies of 2754, 3332, 3551 and 3788 Hz , These figures indicate 
that the shell vibrates predominantly along the large flat sides of the compressor 
at points where the suspension springs are attached to the shell wall at these 
natural frequencies. When referred to figure 1 of the noise radiation this 
frequency range is also the range where the noise radiates efficiently from the 
large flat sides of the compressor. Thus, there is good correlation between the 
acoustic data and structural data for these frequencies. The slight discrepancies 
in the structural natural frequencies and the acoustic data. Acoustic data have 
been picked up at the shell temperature of the reciprocating piston compressor 
that could reach up to 110 C. 




















































































The modal analysis results also indicate that the compressor suspension springs 
are attached to a point on the shell where the shell is comparatively compliant. 
Thus, the vibrational energy transmitted through the springs to the compressor 
shell can and did effectively excite the shell vibrations. Also, significant shell 
vibrations occur along the large flat sides of the compressor shell indicating the 
curvature of the shell needs to be increased to add stifihiess to the shell. 

Based on the results of the shell modal analysis, it is recommended the 
suspension springs moved away from the compliant side walls of the shell. A 
four spring arrangement at the bottom of the shell near corners where the 
curvature is sharp would reduce the amount of vibration energy transferred to 
the shell because of the reduced input mobility of the shell at these locations. 

It is also believed increasing the stiffness of the shell by increasing the curvature 
will provide noise reduction benefits. The greater shell stiffness lowers the 
amplitude of the shell vibrations. Figure 9, illustrate the third octave change in 
compressor noise with the same compressor in the new shell. An over all noise 
level of 5 dBA has been obtained. 



Figure 9 Compressor noise level improvement after the shell modification. 

The increased shell stiffness also raises the natural frequencies of the shell 
where there is less energy for transfer function response. However, there is a 
possible disadvantage to increasing stiffness of the shell. The higher natural 
frequency lowers critical frequency of the shell thus reducing transmission loss 
of the shell. 


1081 



















Damping treatments can also have obvious benefits in vibration and noise 
reduction. Visco elastic and Acoustic dampings are considered to avoid the 
shell excitations. The application of dampers can also provide up to 5 dBA 
reductions when appropriately located on the shell. 

CONCLUSION 

The results of studies indicate that structural resonances of the shell are indeed 
amplifying the noise due to the pumping harmonics of the reciprocating 
refrigeration compressor to cause significant noise radiation outside of 
compressor. 

In order to tackle with this noise problem, within the scope of these studies 
two different effective shell noise control are considered based on the results of 
numerical and experimental structural analysis and acoustic features of the 
reciprocating compressor. First, shell noise control method is the redesign of 
the shell with increased stiffness by replacing all the abrupt changes in the 
curvature with the smooth continuous changes. It is apparent that change in the 
shell configuration can shift the first shell resonance from 1750 Hz to nearly 
3200 Hz. The results of the redesign effects can reach up to 3-5 dBA reduction 
on third octave noise levels. Second treatment that could be applied against the 
excitation of shell resonances are considered as the acoustic and viscoelastic 
dampers. These dampers can be chosen to operate efficiently at the shell 
resonant frequencies. These two applications can also provide up to 2-5 dBA 
reduction on the third octave band of the noise emission but the long term 
endurance and temperature dependence of these materials can always be a 
question mark when considered from the manufacturer point of view. 


REFERENCES 

1. JFROBATTAand ID. JONES 1991, Purdue University, School of 
Mechanical Engineering, The Ray Herrick Laboratories, Report no: 1912-1 
HL 91-9P, 73-84, Investigation of Noise Generation Mechanisms and 
Transmission Paths of Fractional Horsepower Reciprocating Piston and 
Rolling Piston Compressors 

2. HAMILTON IF 1988, Purdue University, School of Mechanical 
Engineering, The Ray Herrick Laboratories, 207-213 Measurement and 
Control of Compressor Noise 

3. C OZTURK, A AQIKGOZ and J L MIGEOT 1996, International 
Compressor Engineering Conference at Purdue, Conference Proeceeding, 
Volume II, 697-703, Radiation Analysis of the Reciprocating Refrigeration 
Compressor Casing 


1082 


A COMPARATIVE STUDY OF MOVING FORCE IDENTIFICATION 


T.H.T. Chan, S.S. Law, T.H. Yung 

Department of Civil & Structural Engineering, 

The Hong Kong Polytechnic University, Hung Horn, Kowloon, Hong Kong 


ABSTRACT 

Traditional ways to acquire truck axle and gross weight information are 
expensive and subject to bias, and this has led to the development of Weigh- 
in-Motion (WIM) techniques. Most of the existing WIM systems have been 
developed to measure only the static axle loads. However dynamic axle loads 
are also important. Some systems use instrumented vehicles to measure 
dynamic axle loads, but are subject to bias. These all prompt the need to 
develop a system to measure the dynamic axle loads using an unbiased 
random sample of vehicles. This paper aims to introduce four methods in 
determining such dynamic forces from bridge responses. The four methods are 
compared with one another based on maximum number of forces to be 
identified, minimum number of sensors, sensitivity towards noise and the 
computation time. It is concluded that acceptable estimates could be obtained 
by all the four methods. Further work includes merging the four methods into 
a Moving Force Identification System (MFIS). 


INTRODUCTION 

The truck axle and gross weight information have application in areas 
such as the structural and maintenance requirements of bridges and pavements. 
However, the traditional ways to acquire that are expensive and subject to bias, 
and this has led to the development of Weigh-in-Motion (WIM) techniques. 
Some systems are road-surface systems which make use of piezo-electric 
(pressure electricity) or capacitive properties to develop a plastic mat or 
capacitive sensors to measure axle weight [1]. Another kind of WIM system is 
the under-structure systems in which sensors are installed under a bridge or a 
culvert and the axle loads are computed from the measured responses e.g. 
AXWAY [2] and CULWAY [3]. All the above mentioned systems can only 
give the equivalent static axle loads. However dynamic axle loads are also 
important as they may increase road surface damage by a factor of 2 to 4 over 
that caused by static loads [4]. Some systems use instrumented vehicles to 
measure dynamic axle loads [5], but are subject to bias. These all prompt the 
need to develop a system to measure the dynamic axle loads using unbiased 
random samples of vehicles. Four methods are developed to determine such 


1083 


dynamic forces from bridge responses which include bending moments or 
accelerations. 


EQUATION OF MOTIONS FOR MOVING LOADS 

The moving force identification methods described in this paper are the 
inverse problems of an predictive analysis which is defined by 0‘Connor and 
Chan [6] as an analysis to simulate the structural response caused by a set of 
time-varying forces running across a bridge. Two models can be used for this 
kind of analysis. 


A. Beam-Elements Model 


0‘Connor and Chan [6] model the bridge as an assembly of lumped 
masses interconnected by massless elastic beam elements as shown in Figure 1, 
and the nodal responses for displacement or bending moments at any instant 
are given by Equations (1) and (2) respectively. 


Moving Loads 





... ip. Ip, ^ 

-D—D—2^ 


Lumped Masses 1 2 ... N-1 N 


Figure 1 - Beam-Elements Model 


{Y} = [Y,]{F}-lY,][Am]{n-m[C]{Y} ( 1 ) 

{5M} = [5MJ{P}-[5M;][Am]{y}-[5MJ[C]{f} (2) 

where [P] is a vector of wheel loads, [Am] is a diagonal matrix containing 

values of lumped mass, [C] is the damping matrix, 5M,L,7,7are the nodal 
bending moments, displacements, velocities and accelerations respectively, 
[R,^] {'R ’ can be Y or BAP) is an m x n matrix with the ith column representing 
the nodal responses caused by a unit load acting at the position of the ith wheel 
load and [i?J {'R' can be Y or BM) is an n x n matrix with the ith column 
representing the nodal responses caused by a unit load acting at the position of 
ith internal node. 


1084 


B. Continuous Beam Model 


Assuming the beam is of constant cross-section with constant mass per 
unit length, having linear, viscous proportional damping and with small 
deflections, neglecting the effects of shear deformation and rotary inertia 
(Bernoulli-Euler’s beam), and the force is moving from left to right at a 
constant speed c, as shown in Figure 2, then the equation of motion can be 
written as 


P 


d-v{xj) 

a- 


a 


+ EI 


^v(x,0 

a- 


S(x-ct)f{t) 


(3) 


where v(x,0 is the beam deflection at point x and time t; p is mass per unit 
length; C is viscous damping parameter; £ is Young's modulus of material; I 
is the second moment of inertia of the beam cross-section; f(t) is the time- 
varying force moving at a constant speed of c, and Sft) is Dirac delta function. 


\/(0 



—O 


Figure 2 - Simply supported beam subjected to a moving force f(t) 

Based on modal superposition, the dynamic deflection y(x,t) can be 
described as follows; 


v{x,t) = Y,<i'„{x)q„{t) 

}}=\ 


(4) 


where n is the mode number; 0n(x^) is the mode shape function of the n-th 
mode and qyi(t) is the n-th modal amplitudes. 

Based on the above mentioned predictive analyses, four Moving Force 
Identification Methods (MFIM) are developed. 


1085 


FIRST MOVING FORCE IDENTIFICATION METHOD 
INTERPRETIVE METHOD I - BEAM-ELEMENTS MODEL (IMI) 


It is an inverse problem of the predictive analysis using beam-elements 
model From Equation (1), it can be seen that if Y is known at all times for all 
interior nodes, then 7 and Y can be obtained by numerical differentiation. 
Equation (1) becomes an overdetermined set of linear simultaneous equations 
in which the P may be solved for them. However a particular difficulty arises 
if measured BM are used as input data. Remembering that the moving loads P 
are not normally at the nodes, the relation between nodal displacements and 
nodal bending moments is 

{r}^[Y,]{BM} + [Yc]{B} ( 5 ) 

where allows for the deflections due to the additional triangular 

bending moment diagrams that occur within elements carrying one or more 
point loads P. [}^]can be calculated from the known locations of the loads. 
[}^^] and {BM}aic known, but {7} cannot be determined without a 
knowledge of (P}. 0‘Connor and Chan [6] describe a solution uses values of 
{P} assumed from the previous time steps. 


SECOND MOVING FORCE IDENTIFICATION METHOD 
INTERPRETIVE METHOD II - CONTINUOUS BEAM MODEL (IMII) 

From the predictive analysis using continuous beam model, if the ith- 

iTTX 

mode shape function of the simply supported Euler‘s beam is sin-—-, then 
the solution of Equation (3) takes the form 

v = ^sin—F;(0 (6) 

/=i L 

where V.(t ), (i = 1, 2, • • •) are the modal displacements. 

Substitute Equation (6) into Equation (3), and multiply each term of 
Equation (3) by the mode shape function sin(y;r.x / L ), and then integrate the 
resultant equation with respect to x between 0 and L and use the boundary 
conditions and the properties of Dirac function. Consequently, the following 
equation can be obtained 


1086 


op 


u 2 El ^ 

where = — 5 —- C(y, = 


C 




2/rryy, 


at the j-th mode. 


If there are more than one moving loads on the beam, Equation (7) can be 
written as 








. 7r(ct-Xk) 

._7r(ct~x,) 

7r{ct-x^) ■ 


■^.1 


'2CM' 


'cojv; 


sin-^ 

L 

sin-^ • 

L 

sin- — 

L 

'K 

V, 


2C,co,K 

+ 

CO IVj 

_ 2 

. 27r(ct-Xt) 
sin 

. 27r(ct~x^} 
sm-=- ■ 

. 27r{ct - x .) 

•• sm—- — 

Pi 




~ Ml 

A 

.1 





_2C,.a)/„_ 


_coX_ 


. n7r(ct-x,) 
sin—^^ 

. riKict-x^) 
sin—^- — . 

. n7c{ct-x^) 

•• sin—- — 

A. 







L 

L 

L 



( 8 ) 


in which is the distance between the k-th load and the first load and x, = 0 . 


If Pi Pk are constants, the closed form solution of Equation (3) is 

/ N ^ 1 • ■ j7E{ct-Xi) a . . . , ^ {Q\ 

48E/f:i' L \ L j ^ J 

nc 


in which a = 


Lcd 


If we know the displacements of the beam at x,, x,, • • •, , the moving 

loads on the beam are given by 

{v} = [S,„.]{/>} (10) 

in which {v}=[v| v, ■■■ v,]' {•^’} = [-^1 ^2 ■" 






^n- 


, where 


L" A 1 . j7rx,J . J7r{ct-x.) a . , . , , 

= 7577X^2772- 27 ^““ 7 ~ -7-/c) 

48£:/^;-0 --a ) L \ L j 


1087 


If l>k, that means the number of nodal displacements is larger than or 
equal to the number of axle loads, then according to the least squares method, 
the equivalent static axle load can be given by 

{/>}=([s,,]''[s„])''[s,q^{v} (11) 

If the loads are not constant with time, then central difference is used to 
proceed from modal displacements to modal velocities and accelerations. 
Equation (8) becomes a set of linear equations in which P^. for any instant can 
be solved by least squares method. Similar sets of equations could be obtained 
for using bending moments to identify the moving loads. 


THIRD MOVING FORCE IDENTIFICATION METHOD 
TIME DOMAIN METHOD (TDM) 


This method is based on the system identification theory [7]. Substituting 
Equation (4) into Equation (3), and multiplying each term by 0j(x), integrating 
with respect to x between 0 and L, and applying the orthogonality conditions, 
then 


dt- 




dt 


M., 


( 12 ) 


where con is the modal frequency of the n-th mode; is the damping ratio of 
the n-th mode; Mn is the modal mass of the n-th modt, pn(t) is the modal force 
and the mode shape function can be assumed as 0„(x) = sin(^;w / L ). 

Equation (12) can be solved in the time domain by the convolution 
integral, and yields 

= ( 13 ) 

^11 0 

where ~ sin(^u„0, t >0 (14) 

and co\, (15) 

Substituting Equation (13) into Equation (4), the dynamic deflection of 
the beam at point x and time t can be found as 

v(x,0 = y—^sin^^ sin6>|,(^ - '^)sin^^^^/(T)dr (16) 

fxpLo),, L i L 


1088 


A. Force Identification from Bending Moments 


The bending moment of the beam at point x and time t is 


m{xj) = -El- 


3c- 


(17) 


Substituting Equation (16) into Equation (17), and assuming the force f(t) 
is a step function in a small time interval and f(t) =0 at the entry and exit, then 
let 


^ 2El7r' n~ . yitu: ^ 

C,„ = —73 -^sm—Ar, 

pi co„ L 

{k) = sin(<:y'„ A/A:), 

^ . ,n7(cNi 

52(*) = sm(—^/c) 


(18) 


(19) 


N„=- 

Equation (17) can be expressed as 


cEt 


m(2)] 



0 

... 0 ' 

/(i) 

m(3) 

< 

■=S c,„ 
/) = 1 

E;,S,{2)S,i\) 

e:S,{\)S,{2) 

... 0 

/(2) 

m{N) 


_E::"S,iN-\)S,il) 

E;^--S,{N~2)S,i2) 

Kc. 



( 20 ) 


where A^ is the sample interval and N+I is the number of sample points, and 


f>„=£r'''“*'S,(Al-A^„+l)S,(//„-l) 


Equation (20) can be simplified as 

B f = m ( 21 ) 

(A'-l)x(;V/^-|) ('V/^->)xl (,V_l)xl 

U N = matrix B is a lower triangular matrix. We can directly find 
the force vector f by solving Equation (21). If 77 > and/or 77/ bending 
moments (77/ > 1) are measured, least squares method can be used to find 

the force vector f from 


1089 


( 22 ) 


B, ■ 


m, 


f 

m. 

®-v,. 




The above procedure is derived for single force identification. Equation 
(21) can be modified for two-forces identification using the linear 
superposition principle as 

“B. 0 

B, B, 

B. B, 

where B 3 [N^x (Nb-1)] , Bt [(N -1-2 NJx (Ng-l)], and B^ [NjX (Nb -1)] are 
sub-matrices of matrix B. The first row of sub-matrices in the first matrix 
describes the state having the first force on beam after its entry. The second 
and third rows of sub-matrices describe the states having two-forces on beam 
and one force on beam after the exit of the first force. 

B. Identification from Bending Moments and Accelerations 



Similarly the acceleration response of the beam can be expressed as 

A f = V 


(24) 


The force can also be found from the measured acceleration from 
Equation (24). If the bending moments and accelerations responses are 
measured at the same time, both of them can be used together to identify the 
moving force. The vector m in Equation (21) and v in Equation (24) should be 
scaled to have dimensionless unit, and the two equations are then combined 
together to give 


A/iv||J [v/llvlj 


(25) 


where Ihli is the norm of the vector. 


FOURTH MOVING FORCE IDENTIFICATION METHOD 
FREQUENCY DOMAIN METHOD (FDM) 

Equation (12) can also be solved in the Frequency Domain. Performing 
the Fourier Transform for Equation (12), 


1090 


1 


1 


(26) 


where 




co;, -CO- +24„co„co M„ 


W 


i 

—00 


(27) 

(28) 


Let 




_1_ 

co;,-co- +2^„a„a 


(29) 


Hn(co) is the frequency response function of the n-th mode. Performing the 
Fourier Transform of Equation (4), and substituting Equations (26) and (29) 
into the resultant equation, the Fourier Transform of the dynamic deflection 
v(x,t) is obtained as 




A. Force Identification from Accelerations 

Based on Equation (30), the Fourier Transform of the acceleration of the 
beam at point .x and time t can be written as 

V(x,a) = -®^X^(D„(x)//„(cy)P„(cy) (31) 

Considering the periodic property of the Discrete Fourier Transform (DFT), 
and let 




A/^m- 




Equation (32) can be rewritten as 


(32) 


1091 



,m = 0,l, (33) 


V{m) = X H„(m)'i'„(m)[F,{0) + iF,(0)] 

/;=1 

A^/2-1 « _ 
k=\ H=t 

Nn~\ =0 _ 

/t = l /) = l 

+E /2)[F„(A^ /2) - iivCA^ / 2)] 

«=I 

where is the Fourier Transform of the n-th mode shape, and F is the 
Fourier Transform of the moving force. 

Writing Equation (33) into matrix form and dividing F and V into real and 
imaginary parts, it yields 



Because F, (0) = 0, Fj (N / 2) = 0,Vj (0) = 0,F; (/ 2) = 0, Equation (34) can 
be condensed into a set of N order simultaneously equations as 

(35) 

Fr and F, can be found from Equation (35) by solving the Nth order linear 
equation. The time history of the moving force f(t) can then be obtained by 
performing the inverse Fourier Transformation. 

If the DFTs are expressed in matrix form, the Fourier Transform of the 
force vector f can be written as follows if the terms in f are real [8]. 

F = —Wf (36) 

N 

whereW = e''^‘‘^^^ (37) 


1092 



0 0 0 0 0 

0 1 2 ••• -2 -1 

0 2 4 ••• -4 -2 

0 -2 -4 ••• 4 2 

0 -1 -2 2 1 


yv X 


The matrix W is an unitary matrix, which means 

w-‘ =(w* y 


(38) 


where W* is a conjugate of W. Substituting Equation (36) into Equation (35), 


V = —Al 

N 


W 0 


-1 

X 

_Ij 



(39) 


or 


V=^ A fe 


(40) 


linking the Fourier Transform of acceleration V with the force vector fg of 
the moving forces in the time domain. Wg is the sub-matrix of W. If N = 
fg can be found by solving the Mh order linear equations. If N> or more 
than one accelerations are measured, the least squares method can be used to 
find the time history of the moving force f(t). 

Equation (40) can be rewritten as follows 



(41) 


relating the accelerations and force vectors in the time domain. Also if N - 
Nb, fe can be found by solving the Nth order linear equation. If N > Nb or 
more than one acceleration are measured, the least squares method can be 
used to find the time history of the moving force f(t). 


If only Nc {Nc ^ N) response data points of the beam are used, the 
equations for these data points in Equation (41) are extracted, and described as 


V, =(w*)" A 

C V B / A/x/V 


c 


B 

Nr X N 


W 3 

/Vx;V, 


yV„x! 


(42) 


1093 


In usual cases Nc > Ng, so the least squares method is used to find the time 
history of the moving force f(t). More than one acceleration measurements at 
different locations can be used together to identify a single moving force for 
higher accuracy. 


B. Identification from Bending Moments and Accelerations 


Similarly, the relationships between bending moment m (and M) and 
the moving force f can be described as follows, 

M = ^ B W is (43) 

/Vxl N N^Nn 



(44) 

(45) 


The force vector fg can be obtained from the above three sets of 
equations. Furthermore, these equations can be combined with Equations (40), 
(41) and (42) to construct overdetermined equations before the equations are 
scaled. Two forces identification are developed using the similar procedure as 
that for the Time Domain Method. 


COMPARATIVE STUDY 

The first moving force identification method is implemented in a 
computer program using FORTRAN, while the other three methods are 
implemented under the environment of a high performance numerical 
computation and visualization software. The predictive analysis using beam- 
elements model is used to generate the theoretical bridge responses and the 
four moving force identification methods then use these responses to recover 
the original dynamic loads. In this study, if at least 80% of the identified 
forces at any instant of any load lie within ± 10% of the original input force, 
the method is considered acceptable. It is found that all the four methods can 
give acceptable results. 

It is decided to carry out a preliminary comparative study on the four 
methods in order to study the merits and limitations of each method so as to 
consider the future development of each method and devise a plan to develop a 


1094 


moving force identification system which can make use of the benefits of all 
the four methods. 


A. Maximum Number of Forces 


This is to examine the maximum number of axle loads that can be 
identified by each method. Theoretically, provided that sufficient number of 
nodal sensors are installed, IMI and IMII can be used to identify as many loads 
as the system allows. Basically, the number of axle loads cannot be larger than 
the number of nodal sensors. Regarding TDM and FDM, as the formulation of 
the governing equation is derived for two moving forces, the maximum 
number of axle loads that can be identified is two. 

B. Minimum Number of Sensors 


Based on a study of common axle spacings of vehicles currently 
operating on Australian roads, and the cases with zero nodal responses, 
0‘Connor and Chan [6] state the relationships of the minimum number of 
sensors used for IMI and the span length of a bridge as follows: 


Using bending moment, for span length L > 4.8m, 

Min. number of nodal moments required = int{ 
Using displacements, for span length L > 13.8m, 

Min. number of nodal displacement required = INj{ 
and for span length L <13.8m, 

Min. number of nodal displacement required 


L - 4.8 ' 
1.7 

L-UX 




3.7 


+ 4 


+ 6 




(46) 

(47) 

(48) 


For IMII, it is found that the number of sensors required are generally less 
than that for IMI. Regarding TDM and FDM, the programs are not as flexible 
as that for IMI and IMII and it is not easy to change the number of sensors. 
Meanwhile the sensors are fixed to be at 1/4, 1/2 and 3/4 of the span. 

C. Sensitivity towards Noise 


In general, all the four methods can compute the identified forces exactly 
the same as those given to the predictive analysis to generate the 
corresponding responses. It is decided to add white noise to the calculated 
responses to simulate polluted measurements and to check their sensitivity 
towards noise. The polluted measurements are generated by the following 


1095 


equations: 


m = n>o.nkui«.«] x N„, 


(49) 


where Ep is a specified error level; is a standard normal distribution 
vector (with zero mean value and unity standard deviation). 

Several cases are studied using Ep =1%, 3%, 5% and 10%. It is found 
that when using bending moments for IMI and IMII, and if Ep is less than 3%, 
acceptable results can be obtained. For noise which is greater than 3%, a 
smoothing scheme should be adopted to smooth the simulated data. 
Acceptable results cannot be obtained for Ep > 10%. Besides, both IMI and 
IMII cannot give acceptable results when using displacements. 

Both TDM and FDM cannot give acceptable results when using 
displacements only, accelerations only or bending moments only. In general 
TDM and FDM are less sensitive to noise when comparing to IMI and IMII. 
They can give acceptable results for Ep up to 5 % without any smoothing of 
the polluted simulated data. 

D. Computation Time 

In general, the computer program for IMI only takes few seconds to 
identify moving forces. In order to compare the computation time, IMI is 
implemented in the same environment as the other three methods. It is found 
that IMI and IMII take about 2-3 minutes to give the identified forces for a 
case of two axle loads using a 80486 computer. However, under the same 
working conditions, TDM and FDM almost take a whole day for any one of 
them to identify two moving forces. It is due to the fact that both of them 
require to set up an huge parametric matrix. 


CONCLUSIONS 

Four methods are developed to identify moving time-varying force and 
they all can produce acceptable results. From a preliminary comparative study 
of the methods, it is found that IMI and IMII have a wider applicability as the 
locations of sensors are not fixed and it can identify more than two moving 
forces. However, TDM and FDM are less sensitive to noise and require less 
number of sensors. It is decided to further improve the four methods and then 
a more detailed and systematic comparison can be carried out afterwards. The 
possible development of the methods are described as follows. 


1096 


Both the IMI and IMII are developed to work with one kind of responses, 
e.g. either displacements or bending moments. It is suggested to modify the 
programs to use mixed input parameter, e.g. use bending moments as well as 
accelerations as that for TDM or FDM. Regarding the TDM and FDM, as the 
basic formulations are based on two-axle moving forces, so it is necessary to 
modify the governing equations for multi-axle. In addition, the computation 
time for TDM or FDM under the environment of the high performance 
numerical computation and visualization software used is unbearable. It is 
expected that the time will be significantly reduced if the methods are 
implemented in programs using standard programming languages like 
FORTRAN 90 or C. Then the four methods can be combined together and 
merged into a Moving Force Identification System (MFIS) so that it can 
automatically select the best solution routines for the identification. 


ACKNOWLEDGMENT 

The present project is funded by the Hong Kong Research Grants Council. 


REFERENCES 

1. Davis, P. and Sommerville, F., Low-Cost Axle Load Determination, 
Proceedings, 13th ARRB & 5th REAAA Combined Conference, 1986, 
Part 6, p 142-149. 

2. Peters, R.J., AXWAY - a System to Obtain Vehicle Axle Weights, 
Proceedings, 12th ARRB Conference, 1984, 12 (2), p 10-18. 

3. Peters, R.J., CULWAY - an Unmanned and Undetectable Highway Speed 
Vehicle Weighing System, Proceedings, 13th ARRB & 5th REAAA 
Combined Conference, 1986, Part 6, p 70-83. 

4. Cebon, D. Assessment of the Dynamic Wheel Forces Generated by 
Heavy Vehicle Road Vehicles. Symposium on Heavy Vehicle Suspension 
Characteristics, ARRB, 1987. 

5. Cantineni, R., Dynamic Behaviour of Highway Bridges Under The 
Passage of Heavy Vehicles. Swiss Federal Laboratories for Materials 
Testing and Research (EMPA) Report No. 220, 1992, 240p. 

6. O'Connor, C. and Chan, T.H.T., Dynamic Wheel Loads from Bridge 
Strains. Structural Engineering ASCE, 114 (STS), 1988, p. 1703-1723. 

7. Briggs, J.C. and Tse, M.K. Impact Force Identification using Extracted 
Modal Parameters and Pattern Matching, International Journal of Impact 
Engineering, 1992, Vol. 12, p361-372. 

8. Bendat, J.S. and Piersol, J.S., Engineering Application of Correlation and 
Spectral Analysis. John Wiley & Sons, Inc. Second Edition, 1993. 


1097 


1098 


ESTIMATING THE BEHAVIOUR OF A 
NONLINEAR EXPERIMENTAL MULTI DEGREE 
OF FREEDOM SYSTEM USING A FORCE 
APPROPRIATION APPROACH 


P.A. Atkins J.R.Wright 

Dynamics and Control Research Group 
School of Engineering, Simon Building, University of Manchester, 
Oxford Road, Manchester. M13 9PL 


ABSTRACT 

The identification of nonlinear multi degree of freedom systems involves a 
significant number of nonlinear cross coupling terms, whether the identifi¬ 
cation is carried out in spatial or modal domains. One possible approach 
to reducing the order of each identification required is to use a suitable 
pattern of forces to drive any mode of interest. For a linear system, the 
force pattern required to drive a single mode is derived using a Force Ap¬ 
propriation method. This paper presents a method for determining the 
force pattern necessary to drive a mode of interest of a nonlinear system 
into the nonlinear region whilst the response is controlled to remain in pro¬ 
portion to the linear mode shape. Such an approach then allows the direct 
nonlinear modal terms for that mode to be identified using the Restoring 
Force method. The method for determining the relevant force patterns is 
discussed. The implementation of the method for experimental systems is 
considered and experimental results from a two degree of freedom ’bench¬ 
mark structure’ are presented. 


INTRODUCTION 

Force Appropriation [1] is used in the analysis of linear systems to de¬ 
termine the force patterns which will induce single mode behaviour when 
applied at the relevant natural frequency. This technique is used in the 
aerospace industry during Ground Vibration Tests: each normal mode of 


1099 


a structure is excited using the derived force pattern and thus identified 
in isolation. Current practice, when the presence of nonlinearity is sus¬ 
pected, is to increase input force levels and monitor the variation of tuned 
frequencies. Some information about the type of nonlinearity present may 
be found, but no analytical model can be derived. Thus predictions for 
behaviour at higher levels of excitation axe not possible. 

A number of techniques for identifying nonlinearity, for example the Restor¬ 
ing Force method [2], have been demonstrated on systems with low num¬ 
bers of degrees of freedom. Unfortunately, in practice, structures have a 
large number of degrees of freedom, often with a high modal density. A 
classical Restoring Force approach to the identification of such systems 
could involve a prohibitive number of cross coupling terms. The ability to 
treat each mode separately, by eliminating the effects of the cross coupling 
terms, would thus.be advantageous. Subsequent tests could then evaluate 
the cross coupling terms. 

For these reasons it would be useful to extend Force Appropriation to the 
identification of nonlinear systems. An approach has been developed [6] 
that allows an input force pattern to be derived that will result in a non¬ 
linear response in the linear mode shape of interest. This force pattern is 
derived using an optimisation approach. The mode of interest can then be 
identified using a single degree of freedom nonlinear identification method. 
In this work the Restoring Force method is used to examine the nonlinear 
response of a particular linear mode and an application of this approach 
to a two degree of freedom experimental system is presented. 


THEORY 


The theoretical approach is demonstrated for the two degree of freedom 
system with spring grounded nonlinearity shown in figure 1. The equations 
of motion for this system in physical space are: 


m 0 
0 m 


+ 


+ 


(1 + a)c 

—ac 


'±1 1 

—ac 

(1 + d)c 

\ 

X2 J 

(l -}- Q^k 

—ak 


fill 

—ak 

(1 + a)k 






l^x\ 

0 



where is the cubic stifEhess coefficient and a is a constant that allows the 
frequency spacing of the natural frequencies to be varied. These equations 
can be transformed to linear modal space using the transformation: 


{i} = [ij>]{u} (2) 

where [(j>] is the modal matrix of the underlying linear system and the 
vector {u} defines the modal displacements. For this symmetrical system 


1100 


the modal matrix is 




1 1 
1 -1 


(3) 


The equations of motion transformed to linear modal space using the nor¬ 
malised modal matrix are: 


771 0 \ ill 

0 771 (112 


+ 


c 0 


0 (1 + 2g)c 

1 ^ 2 / 

■jb 0 

Ull 

0 (1 + 2a)k 

1 ^ 2 ] 


P{Ul-U2y/4:' 

-U2)V4 



where {p} is the modal input vector. It can be seen from the above equa¬ 
tions that the cubic nonlinearity couples the modes in linear mod^ space; 
in fact there are a significant number of terms for a single nonlinearity. 
The proposed method aims to determine the force pattern that will reduce 
the response of this system to that of a single mode. 

It was shown in a previous paper [3] that this can be achieved by seeking a 
force vector that will cause motion only in the target mode, by eliminating 
motion in the coupled mode. In practice, physical data from transducers 
are available. Any subsequent transformations would be time consuming. 
It is shown below that causing motion in one mode to be zero is equiva¬ 
lent to forcing motion in a linear mode shape, mode one in this example. 
Consider the coordinate transformation {a;} = [<^]{ii} or more explicitly 
for the two degree of freedom system in Figure 1: 


Thus 




1 

2 


‘l -1 
1 1 



(5) 

( 6 ) 


and enforcing the first mode shape (1,1) in physical space should give a 
second modal displacement of zero. 


1 

2 


1 -1 
1 1 


1 

1 



(7) 


So if an excitation is applied which causes the nonlinear system to vibrate 
in its first linear mode shape, the response will be composed only of ui and 
the influence of the coupled mode, U 2 , will then have been eliminated. 

The method must therefore derive a force pattern which will cause the 
system to vibrate in one of its linear mode shapes. It has been shown 


1101 


in a previous paper [3] that if the response contains harmonics then the 
force pattern must also contain harmonics in order to control the harmonic 
content of the response. In theory, the responses will be an infinite series 
of harmonics, but this series is truncated in this case of a cubic stiffness 
nonlinearity to include only the fundamental and third harmonic terms. 

The physical input forces will thus be of the form: 

fi(t) = Fii cos(a;ea;t 4- ^ii) 4- F 13 cos(a;ea;t + (^ 13 ) ( 8 ) 

f2{t) = F21 C0s(a;ea;t + (j>2l) 4* F23 COS{uJext + fe) (9) 

where is the excitation frequency. Parameters for these force patterns 
may then be chosen such that only mode one is excited. 


OPTIMISATION APPROACH 

In general, no a priori model of the system exists so an optimisation routine 
is used to determine the force pattern parameters required to maximise the 
contribution of the mode of interest. The objective function, the quantity 
that the optimisation routine seeks to minimise, must be representative 
of the deviation of the response from the target linear mode shape. The 
objective function, F, that was chosen in this case was based on the vector 
norm [4] of the two physical responses, Xi and X 2 , and is shown below: 



where and 02 are elements of the mode shape vector for the target 
mode. This summation is carried out over one cycle of the fundamental 
response. The number of data points per cycle is npts and Xki the kth 
response at the itk sample. This objective function allows the response 
to contain harmonics and can be extended to more degrees of freedom by 
choosing a reference displacement and subtracting further displacements 
from it. The Variable Metric optimisation method [5] was used in this 
work as it has been found to produce the best results for simulated data. 
The application of this method to a two degree of freedom system such 
as that shown in figure 1 is detailed in [6]. Optimised force patterns are 
obtained at several levels of input amplitude. These force patterns are then 
applied and the Restoring Force method is used to curve fit the resulting 
modal displacement and velocity time histories to give the direct linear and 
nonlinear coefficients for the mode of interest. 


1102 



EXPERIMENTAL IMPLEMENTATION 


The simulated application of this method assumed that certain parame¬ 
ters were known. In order to carry out an identification of an experimental 
structure, these parameters must be measured or calculated. Some pro¬ 
cessing of experimental data is necessary in order to apply the Restoring 
Force method. The restoring force of a system can be expressed for a single 
degree of freedom system as: 

h{x, x) = f(t) - mx (11) 

where h{x, x) is the restoring force and f{t) the input force. A similar ex¬ 
pression applies to the modal restoring force for an isolated mode. Thus the 
input force, acceleration, velocity and displacement must be calculated at 
the each time instant. A similar expression applies to the modal restoring 
force for an isolated mode. In the experimentaJ case it is usual to measure 
acceleration and input force; the remaining two states must therefore be 
obtained by integration of the acceleration time history. Frequency domain 
integration [7] was used for this purpose. High pass filtering was used to 
remove any low frequency noise which can be amplified by this type of 
integration. Several methods have been suggested for estimating modal 
mass, but in this study a method developed by Worden and Tomlinson [8] 
was used. An estimate for the modal mass is obtained and then an error 
term is included in the curve fit which will iteratively yield a more accu¬ 
rate estimate of the mass. Generally the mass value will converge after one 
iteration. 

The objective function used in the simulations was calculated from the 
displacement time histories. In the experimental case, acceleration was 
used rather than displacement as it was considered that using ’raw’ data 
would be quicker and give less opportunity for error. In the simulated 
case, the system parameters were known a priori so the modal matrix 
of the underlying linear system could be calculated. For most types of 
nonlinearity the response of the system at low input force levels will be 
dominated by linear terms. Normal mode tuning [1] was therefore applied 
at low force levels to yield an approximation to the modal matrix of the 
underlying linear system. 

A quality indicator to give some idea of the effectiveness of the optimisa¬ 
tion performed would be advantageous. Results corrupted by background 
noise, for example, could then be discarded. A perfect optimisation will 
occur when the ratio of measured accelerations exactly matches the mode 
shape ratio specified for the mode of interest. Thus a least squares fit of 
the sampled accelerations was carried out over a cycle of the fundamental 
frequency and the percentage error of the measured mode shape to the 


1103 


required mode shape was calculated. This percentage error will indicate 
whether the optimisation has been successful. 

To assess the accuracy of the parameters estimated using this method, 
an identification was carried out using a conventional Restoring Force ap¬ 
proach in physical space. A band limited random excitation was used, and 
the physical data processed and curve fitted. The physical parameters were 
then transformed to modal space. The direct linear and nonlinear param¬ 
eters for modes one and two are shown in table 1. It should be noted that 
although this conventional Restoring Force approach is possible for this 
two degree of freedom system, it will not generally be possible since the 
number of terms in the curve fit increases dramatically when different t 3 q)es 
of nonlinearity and more degrees of freedom are included. It is carried out 
in this case as a means of validating the proposed method. 


EXPERIMENTAL SETUP 

The rig constructed consisted of two masses on thin legs connected in series 
by a linear spring, each mass being driven by a shaker. A cubic nonlin¬ 
earity was introduced between the first mass and ground using a clamped- 
clamped beam attached at the centre which will yield a cubic stiffness for 
large deflections [9], A schematic diagram of the rig is shown in figure 2. 
The force input by each shaker was measured using a force gauge and the 
acceleration of each mass was measured using an accelerometer in the po¬ 
sitions also shown in figure 2, Acceleration and force data were acquired 
using a multiple channel acquisition system, the optimisation routine was 
carried out on line. 


RESULTS 


Normal mode tuning of the rig gave natural frequencies of 20.67 Hz and 
24.27 Hz and a modal matrix of: 


3.87 5.03 
5.52 -3.27 


( 12 ) 


The excitation frequency was chosen to be slightly lower than the natural 
frequency of the mode of interest in order to avoid the problems associated 
with force drop out which are worst at the natural frequency. For each 
mode optimisation was performed at three input force levels, the highest 
level was as high as possible so as to excite the nonlinearity strongly. The 
optimisation routine was carried out using the voltage input into the signal 
generator as the variable. The force input into the structure was measured 


1104 


for use in the Restoring Force identification but was not used in the opti¬ 
misation as it is not directly controllable. The details of the optimisation 
for each force level are presented in table 2. The optimised forces and re¬ 
sulting accelerations for mode one are shown in figure 3 and figure 4. The 
acceleration data for the optimised force patterns were then integrated and 
the modal restoring force for the mode of interest calculated. The initial 
estimate of modal mass for the calculation of the modal restoring force was 
tahen from a previous paper [10] in which the rig was identified using a 
using a physical parameter identification method. The mass was estimated 
in this paper to be 2.62 kg, this physical mass will then be equal to the 
modal mass since the modal matrix was normalised to be orthonormal. 
The restoring force data was then transformed to modal space. The modal 
restoring force surface obtained using optimised force inputs for mode one 
is shown in figure 5 and a stiffness section through this surface is shown in 
figure 6. The restoring force time histories were then curve fitted against 
modal velocity and displacement. 

The estimated parameters for mode one axe shown in table 3. It can be 
seen that they do not compare very well with those estimates obtained 
using the band limited random excitation. It was suspected that at lower 
excitation levels the estimates were being distorted by linear dependence 
[11]. Linear dependence is a problem which occurs when curve fitting a 
harmonic response from a linear system; the equations of motion may be 
identically satisfied by mass and stiffness terms modified by an arbitrary 
constant. This condition is avoided by the harmonic terms introduced into 
the response by nonlinearity. The curve fit was thus repeated using only the 
data obtained from the highest level of excitation; the estimates obtained 
are shown in table 4. It can be seen that the linear parameter estimates 
now agree well with the band limited random results. The estimates for 
the cubic stiffness coefficient do not appear to agree so well. The standard 
deviation on the cubic stiffness derived from the band limited random 
excitation is approximately a third of the value of the parameter itself. The 
uncertainty on this parameter occurs because the nonlinearity is not very 
strongly excited by this type of excitation. A stiffness section through the 
restoring force surface, figure 7, shows little evidence of a cubic stiffness 
component. If a higher level of excitation were possible then a better 
estimate may be achieved. 

The identification w^ repeated for mode two. The restoring force surface 
obtained and a stiffness section through it are shown in figure 8 and figure 9. 
It can be seen from the stiffness section that the nonlinearity is not very 
strongly excited. The estimated direct modal parameters are shown in 
table 5. It can be seen that these results agree quite well with those 
obtained using band limited random excitation. It is considered that the 
discrepancy between the two sets of results, in particular the mass and 


1105 


stiifiiess estimates, is again due to linear dependence. 


CONCLUSIONS 

An extension of the force appropriation method has been proposed for 
nonlinear systems. In this method, an optimisation routine is used to 
determine the force patterns which will excite a single mode nonlinear 
response . The direct linear and nonlinear modal parameters can then be 
estimated from a curve fit of the modal restoring force surface. The method 
was applied to an experimental two degree of freedom system whose modes 
were coupled in linear modal space by a spring grounded nonlinearity. 
A conventional restoring force identification was performed using a band 
limited random signal for comparison. The parameters estimated from the 
single mode responses were found to agree quite well with those from the 
band limited random tests. 


REFERENCES 


1. Holmes P., Advanced Applications of Normal Mode Testing, PhD 
Thesis, University of Manchester 1996. 

2. Hadid M.A. and Wright J.R., Application of Force State Mapping 
to the Identification of Nonlinear Systems Mechanical Systems and 
Signal Processing, 1990, 4(6), 463-482 

3. Atkins P.A., Wright J.R., Worden K., Manson G.M. and Tomlinson 
G.R., Dimensional Reduction for Multi Degree of Freedom Nonlinear 
Systems, International Conference on Identification in Engineering 
Systems 1996, 712-721 

4. Kreyszig E., Advanced Engineering Mathematics, Wiley, 6th Edition 

5. Press W.H., Teukolsky S.A., Vettering W.T. and Flannery B.P., Nu¬ 
merical Recipes in Fortran Cambridge University Press, 6th Edition 

6. Atkins P.A. and Wright J.R., An Extension of Force Appropriation 
for Nonlinear Systems Noise and Vibration Engineering, Proceedings 
of ISMA21(2), 915-926, 1996 

7. Worden K., Data Processing and Experiment Design for the Restor¬ 
ing Force Method, Part I: Integration and Differentiation of Measured 
Time Data, Mechanical Systems and Signal Processing, 4(4) 295-319, 
1990 


1106 


8. Ajjan Al-Hadid M., Identification of Nonlinear Dynamic Systems us¬ 
ing the Force State Mapping Technique, PhD Thesis, Queen Mary 
College, University of London, 1989 

9. Storer D.M., Dynamic Analysis of Nonlinear Structures Using Higher 
Order Frequency Response Function, PhD Thesis, University of Manch¬ 
ester, 1991 

10. Atkins P. and Worden K., Identification of a Multi Degree of Freedom 
Nonlinear System, Proc. of IMAC XV, 1997 


ACKNOWLEDGEMENTS 

This work was supported by E.P.S.R.C. under research grant number 
GR/J48238 at the University of Manchester 


Modal parameter 

Mode one 

Mode two 

k (N/m) 
c (Nm/s) 

/3(N/m^) 
m (kg) 

4.87 X 10^ 
10.11 

3.83 X 10® 
2.60 

6.49 X 10^ 
9.49 

8.90 X 10* 
3.06 


Table 1: Direct modal parameters estimated from curve fit of band limited 
random data 



Low forcing 

Medium forcing 

High forcing 

Fii (Volts) 

1.0 

2.0 

3.0 

F 21 (Volts) 

1.0 

2.0 

3.0 

u)^ (Hz) 

20.0 

20.0 

20.0 

initial mode shape ratio 

0.72 

0.68 

0.65 

final mode shape ratio 

0.70 

0.70 

0.70 

target mode shape ratio 

0.70 

0.70 

0.70 

percentage error 

0.14 

0.03 

0.71 


Table ,2: Details of optimisation for mode one 





Table 3: Direct modal parameters estimated from optimised responses 


Model parameter Estimated parameter 
k (N/m) 4.57 X 10^ 

c (Nm/s) 8.62 

/?(N/m3) 6.81 X 10® 

m (kg) __2^75_ 

Table 4: Direct modal parameters estimated using high force level only 


Model parameter 

Estimated parameter 

k (N/m) 
c (Nm/s) 

m (kg) 

5.37 X 10^ 

10.16 

2.45 X 10® 

2.29 


Table 5: Direct modal parameters estimated for mode two 



Figure 1: Two degree of freedom system 


1108 












0 s 0.40008 


Figure 3: Optimised forces for mode one at a high force level 


















Figure 4: Accelerations responses to optimised forces 



Figure 5: Modal restoring force surface for mode one 






Force (N) 



Figure 7: Stiffness section through modal restoring force derived from ran¬ 
dom excitation for mode one 


1111 




Figure 8: Modal restoring force surface for mode two 



Figure 9: Stiffness section through modal restoring force for mode two 


1112 






POWER FLOW TECHNIQUES II 


THE OPTIMAL DESIGN OF NEAR-PERIODIC STRUCTURES TO 
MINIMISE NOISE AND VIBRATION TRANSMISSION 


R.S. Langley, N.S. Bardell, and P.M. Loasby 
Department of Aeronautics and Astronautics 
University of Southampton 
Southampton SO 17 IBJ, UK 

1. INTRODUCTION 

An engineering structure is said to be of "periodic" construction if a basic 
structural unit is repeated in a regular pattern . A beam which rests on 
regularly spaced supports is one example of a one-dimensional periodic 
structure, while an orthogonally stiffened cylinder is an example of a two- 
dimensional periodic structure. It has long been known that perfectly periodic 
structures have very distinctive vibration properties, in the sense that "pass 
bands" and "stop bands" arise: these are frequency bands over which elastic 
wave motion respectively can and cannot propagate through the structure [1,2]. 
If the excitation frequency lies within a stop band then the structural response 
tends to be localised to the immediate vicinity of the excitation source. 
Conversely, if the excitation frequency lies within a pass band then strong 
vibration transmission can occur, and it is generally the case that the resonant 
frequencies of the structure lie within the pass bands. 

Much recent work has been performed concerning the effect of random 
disorder on a nominally periodic structure (see for example [3-5]). It has been 
found that disorder can lead to localisation of the response even for excitation 
which lies within a pass band, and this reduces the propensity of the structure 
to transmit vibration. This raises the possibility of designing disorder into a 
structure in order to reduce vibration transmission, and this possibility was 
briefly investigated in reference [6] for a one-dimensional periodic waveguide 
which was embedded in an otherwise infinite homogeneous system. The 
present work extends the work reported in reference [6] to the case of a finite 
near-periodic beam system, which more closely resembles the type of 
optimisation problem which is. likely to occur in engineering practice. The 
beam is taken to have N bays, and the design parameters are taken to be the 
individual bay lengths. Both single frequency and band-limited excitation are 
considered, and two objective functions are investigated: (i) the response in a 
bay which is distant from the applied loading (minimisation of vibration 
transmission), and (ii) the maximum response in the structure (minimisation 
of maximum stress levels). In each case the optimal configuration is found by 
employing a quasi-Newton algorithm, and the physical features of the resulting 
design are discussed in order to suggest general design guidelines. 


1113 


2. ANALYTICAL MODEL OF THE NEAR-PERIODIC BEAM 


2.1 Calculation of the Forced Response 

A schematic of an N bay near-periodic beam structure is shown in Figure 1. 
The structure is subjected to dynamic loading, and the aim of the present work 
is to find the optimal design which will minimise a prescribed measure of the 
vibration response. No matter what type of optimisation algorithm is 
employed, this type of study requires repeated computation of the system 
dynamic response as the design parameters are varied, and it is therefore 
important to employ an efficient analysis procedure. In the present work the 
h-p version of the finite element method (FEM) is employed: with this 
approach the structure is modelled as an assembly of elements which have 
both nodal and internal degrees of freedom. Each element has two nodes and 
the nodal degrees of freedom consist of the beam displacement and slope; the 
internal degrees of freedom are generalised coordinates which are associated 
with a hierarchy of shape functions which contribute only to the internal 
displacement field of the element. The internal shape functions used here are 
the K-orthogonal Legendre polynomials of order four onwards - full details of 
the present modelling approach are given in reference [7]. 

For harmonic excitation of frequency co the equations of motion of the 
complete beam structure can be written in the form 

where M and K are the global mass and stiffness matrices (assembled from the 
individual element matrices taking into account the presence of any mass or 
spring elements and allowing for constraints), q contains the system 
generalized coordinates, F is the generalized force vector, and r| is the loss 
factor, which in the present study is taken to be uniform throughout the 
structure. 

Equation (1) can readily be solved to yield the system response q. In the 
present work it is convenient to use the time averaged kinetic and strain 
energies of each of the N bays as a measure of the response - for the nth bay 
these quantities can be written as and say, where 

r,=(£0V4)9;>„?„. (2.3) 

Here and K„ are the mass and stiffness matrices of the nth bay, and q„ is 
the vector of generalized coordinates for this bay. 

Many of the physical features of the forced response of a near-periodic 
structure can be explained in terms of the free vibration behaviour of the 


1114 


associated perfectly periodic structure. The following section outlines how the 
present finite element modelling approach can be used to study the pass bands 
and stop bands exhibited by a perfect periodic structure. 

2.2 Periodic Structure Analysis 

The finite element method described in section 2.1 can be applied to a single 
bay of a perfectly periodic structure to yield an equation of motion in the form 

Dq=F, D=-coW--(l+/r|)ii:, (^’5) 


where the matrix D is referred to as the dynamic stiffness matrix. In order to 
study wave motion through the periodic system it is convenient to partition D, 
q and F as follows 



r 

Du 

ft: 





D= 


D„ 

D,r 



. F= 

0 



Dr, 





F„ 

V V 


( 6 - 8 ) 


where L relates to the coordinates at the left most node, R relates to those at 
the right most node, and / relates to the remaining "internal" coordinates. 
Equations (4-8) can be used to derive the following transfer matrix relation 
between the displacements and forces at the left and right hand nodes 




V / 


V / 


(9,10) 

Equation (9) can now be used to analyze wave motion through the periodic 
system: such motion is governed by Bloch’s Theorem, which states that 
i^L Fi)=exp{-ie-b){qii -F^) where 8 and 5 are known respectively as the phase 
and attenuation constants. A pass band is defined as a frequency band over 
which 6=0, so that wave motion can propagate down the structure without 
attenuation. It follows from equation (9) that 


(T-Ie 


-ie-6' 




( \ 



0 

F, 


io 

V 


V y 


( 11 ) 


SO that 8 and 6 can be computed from the eigenvalues of T, thus enabling the 
pass bands and stop bands to be identified. 


1115 


2.3 Optimisation Procedure 

Equations (l)-(3) enable the forced response of the system to be calculated for 
any prescribed set of system properties. The aim of the present analysis is to 
compute the optimal set of system properties for a prescribed design objective, 
and in order to achieve this equations (l)-(3) are evaluated repeatedly as part 
of an optimisation algorithm. As an example, it might be required to minimise 
the kinetic energy of bay N by changing the various bay lengths. In this case 
equations (l)-(3) provide the route via which the objective function (the kinetic 
energy in bay N) is related to the design parameters (the bay lengths), and the 
optimisation algorithm must adjust the design parameters so as to minimise the 
objective function. The optimisation process has been performed here by using 
the NAg library routine E04JAF [8], which employs a quasi-Newton algorithm. 
This type of algorithm locates a minimum in the objective function, although 
there is no indication whether this minimum is the global minimum or a less 
optimal local minimum. The probability of locating the global minimum can 
be increased significantly by repeated application of the NAg routine using 
random starts, i.e. random initial values of the design parameters. Numerical 
investigations have led to the use of 30 random starts in the present work. 

3. NUMERICAL RESULTS 

3.1 The System Considered 

The foregoing analysis has been applied to a beam of flexural rigidity El, mass 
per unit length m, and loss factor r|=0.015, which rests on A+1 simple 
supports, thus giving an A’-bay near-periodic system. The design parameters 
are taken to be the bay lengths (i.e. the separation of the simple supports), and 
the design is constrained so that the length of any bay lies within the range 
0.9L^,<1.1L, where L, is a reference length. A non-dimensional frequency 
Q is introduced such that 0.=(oL,N{m/Er), and the non-dimensional kinetic and 
strain energies of a bay are defined as T,'-T,^(EI/L^^\F\^) and 
U,'~U„iEI/L,^\F\'^) where F is the applied point load. As discussed in the 
following subsections, two objective functions are considered corresponding 
to minimum vibration transmission and minimum overall response. In all 
cases the excitation consists of a point load applied to the first bay and the 
response is averaged over 11 equally spaced point load locations within the 
bay. For reference, the propagation constants for a periodic system in which 
all the bay lengths are equal to L, are shown in Figure 2 - the present study is 
focused on excitation frequencies which lie in the range 23<f2<61, which 
covers the second stop band and the second pass band of the periodic system. 

3.2 Design for Minimum Vibration Transmission 

In this case the objective function is taken to be the kinetic energy in bay N, 
so that the aim is to minimise the vibration transmitted along the structure. 
Three types of loading are considered; (i) single frequency loading with Q=50, 


1116 


which lies within the second pass band of the ordered structure; (ii) band- 
limited loading with 40<Q<60, which covers the whole of the second pass 
band; (iii) band-limited loading with 23<Q<61, which covers the whole of the 
second stop band and the second pass band. 

Results for the optimal design under single frequency loading are shown in 
Table 1; in all cases it was found that the bay lengths were placed against 
either the upper bound (U=1.1L,) or the lower bound (L=0.9L,), and significant 
reductions in the energy level of bay N were achieved. In this regard it should 
be noted that the dB reduction quoted on Table 1 is defined as -101og(r;v/^;^r) 
where is the kinetic energy in the final bay of the ordered system. The 
optimal designs shown in Table 1 all tend to consist of a bi-periodic structure 
in which the basic unit consists of two bays in the configuration LU. The pass 
bands and stop bands for this configuration are shown in Figure 3, and further, 
for the optimal 12 bay system is shown in Figure 4 over the frequency 
range 0<Q<250. By comparing Figures 3 and 4 it is clear why the selected 
design is optimal - the new bi-periodic system has a stop band centred on the 
specified excitation frequency Q=50. It can be seen from Figure 4 that the 
improvement in the response at the specified frequency 0=50 is accompanied 
by a worsening of the response at some other frequencies. 

Results for the optimal design under band-limited excitation over the range 
40<Q<60 are shown in Table 2. In some cases two results are shown for the 
optimised "Final Energy": in such cases the first result has been obtained by 
forcing each bay length onto either the upper (U) or lower (L) bound, while 
the second result has been obtained by using the NAg optimisation routine. 
If only one result is shown then the two methods yield the same optimal 
design. The "bound" result is easily obtained by computing the response under 
each possible combination of U and L bay lengths - this requires 2^ response 
calculations, which normally takes much less CPU time than the NAg 
optimisation routine. It is clear from Table 2 that the additional improvement 
in the response yielded by the full optimisation routine is minimal for this 
case. The response curve for the 12-bay system is shown in Figure 5, where 
it is clear that a significantly reduced response is achieved over the specified 
frequency range; as would be expected an increase in the response can occur 
at other frequencies. It is interesting to note that most of the optimal designs 
shown in Table 2 lack symmetry - however, it follows from the principle of 
reciprocity that a design which minimises vibration transmission from left to 
right will also minimise transmission from right to left. It should therefore be 
possible to "reverse" the designs without changing the transmitted vibration 
levels. This hypothesis is tested in Figure 6 for a 12 bay structure - the figure 
shows the energy distribution for the optimal design UUULUULLLLLU and 
for the reversed design ULLLLLUULUUU. Although the detailed distribution 
of energy varies between the two designs, the energy levels achieved in bay 


1117 


12 are identical, as expected. 


Results for the optimal design under wide-band excitation 23<Q<61 are shown 
in Table 3, and the response curve for the 12-bay optimised system is shown 
in Figure 7. The form of optimal design achieved is similar to that obtained 
for the narrower excitation band 40<Q<60, although there are detailed 
differences between the two sets of results. In each case there is a tendency 
for a group of lower bound bays (L) to occur in the mid region of the 
structure, and a group of upper bound bays (U) to occur at either end. This 
creates an "impedance mismatch" between the two sets of bays, which 
promotes wave reflection and thus reduces vibration transmission along the 
structure. By comparing Tables 1-3, it is clear that the achievable reduction 
in vibration transmission reduces as the bandwidth of the excitation is 
increased. 

3.3 Design for Minimum ''Maximum” Strain Energy 

In this case the strain energy U„ of each bay is computed and the objective 
function is taken to be the maximum value of U„. As a design objective, this 
procedure can be likened to minimising the maximum stress in the structure. 
As in the previous section the three frequency ranges Q=50, 40<n<60, and 
23<Q<61 are considered, and the present study is limited to systems having 
9,10, 11, or 12 bays; the optimal designs achieved are shown in Table 4. 

Considering the single frequency results (Q.=50) shown in Table 4, it is clear 
that a large dB reduction is achieved only for those systems which have an 
even number of bays; furthermore, the optimal energy obtained has the same 
value (0.0297) in all cases. This can be explained by noting that for an odd 
number of bays the frequency Q=50 lies near to an anti-resonance of the 
ordered structure, whereas a resonance is excited for an even number of bays - 
this feature is illustrated in Figure 8 for the 12 bay structure. The repeated 
occurrence of the optimal energy 0.0297 arises from the fact that the initial bay 
pattern ULLLUUU occurs in all four designs - it has been found that this 
pattern causes a vibration reduction of over 20dB from bay 1 to bay 8, so that 
the response in bay 1 (the maximum response) is insensitive to the nature of 
structure from bay 8 onwards. 

The optimal "bounded" designs arising for band-limited excitation either tend 
to be of the "UL" bi-periodic type or else nearly all the bays are assigned the 
same length. However it should be noted that in all cases the design produced 
by the NAg optimisation routine offers an improvement over the "bounded" 
design, particularly for the wide-band case (23<n<61). It is clear from Table 
4 that the achieved reduction in strain energy reduces as the bandwidth of the 
excitation is increased. 


1118 


4. CONCLUSIONS 


The present work has considered the optimal design of a near-periodic beam 
system to minimise vibration transmission and also maximum stress levels. 
With regard to vibration transmission it has been found that very significant 
reductions in transmission are achievable with relatively minor design changes. 
The optimum design normally involves placing the design parameters (the bay 
lengths) on the permissible bounds, and this means that a simple design search 
routine can be used in preference to a full optimisation algorithm. With regard 
to minimum stress levels, it has been found that the optimal design for wide¬ 
band excitation is not normally a "bounded" design, and thus use of a full 
optimisation algorithm is preferable for this case. For both vibration 
transmission and maximum stress levels, the benefits obtained from an optimal 
design decrease with increasing excitation bandwidth, but nonetheless very 
significant reductions can be obtained for wide-band excitation. 

REFERENCES 

1. S.S. MESTER and H. BENAROYA 1995 Shock and Vibration 2, 69- 
95. Periodic and near-periodic structures. 

2. D.J. MEAD 1996 Journal of Sound and Vibration 190, 495-524. Wave 
propagation in continuous periodic structures: research contributions 
from Southampton 1964-1995. 

3. C.H. HODGES 1982 Journal of Sound and Vibration 82, 411-424. 
Confinement of vibration by structural irregularity. 

4. D. BOUZIT and C. PIERRE 1992 Journal of Vibration and Acoustics 
114, 521-530. Vibration confinement phenomena in disordered, mono- 
coupled, multi-span beams. 

5. R.S. LANGLEY 1996 Journal of Sound and Vibration 189, 421-441. 
The statistics of wave transmission through disordered periodic 
waveguides. 

6. R.S. LANGLEY 1995 Journal of Sound and Vibration 188, 717-743. 
Wave transmission through one-dimensional near periodic structures: 
optimum and random disorder. 

7. N.S. BARDELL, R.S. LANGLEY, J.M. DUNSDON and T. KLEIN 
1996 Journal of Sound and Vibration 197, 427-446. The effect of 
period asymmetry on wave propagation in periodic beams. 

8. ANON 1986 The NAg Fortran Workshop Library Handbook - Release 
1. Oxford: NAg Ltd. 


1119 


TABLE 1 

Optimal design of 1-D beam structure, to minimise energy transmission, D.=50. 


Original Energy; Non-dimensional kinetic energy in bay N of the periodic structure. 
Final Energy; Non-dimensional kinetic energy in bay N of the optimised structure. 


No. of 

Optimal Pattern 

Original 

Final 

Reduction 

Bays, N 


Energy 

Energy 

(dB) 

4 

UULU 

0.276E 1 

0.804E-3 

35.348 


ULULU 

0.609E-1 

0.179E-3 


■S 

UULULULU 

0.674E 0 

0.613E-5 



ULULULULU 

0.564E-1 

0.135E-5 

46.216 

1— 

10 

UULULULULU 

0.424E 0 

0.532E-6 ^ 


11 

ulululululu 

0.535E-1 

0.n7E-6 

56.604 

12 

UULULULULULU 

0.289E 0 

0.461E-7 

67.966 

13 

ULULULULULULU 

0.502E-1 

O.lOlE-7 

66.950 

16 

UULULULULULULULU 

0.154E0 

0.346E-9 
__— 

86.484 

17 

ululululululululu 

0.43 lE-1 

0.761E-10 

87.529 1 


1120 



















TABLE 2 

Optimal design of 1-D beam structure, to minimise energy transmission, 40<Q<60. 

Original Energy: Non-dimensional kinetic energy in bay N of the periodic structure. 

Final Energy: Non-dimensional kinetic energy in bay N of the optimised structure. 


No. of 

Optimal pattern 

Original | 

Final 

Reduction 

Bays, N 


Energy 

Energy 

(dB) 

■■ 

ULLU 

0.670E 0 

0.103E-1 

18.112 

5 

ULLLU 

0.63 IE 0 

0.735E-2 

19.338 




0.711E-2 

19.482 

6 

UULLLU 


0.221E-2 

22.407 

7 

ULLUULU 

0.463E 0 

0.171E-2 

24.335 

8 

UULLLLLU 

0.430E 0 

0.966E-3 

26.487 




0.914E-3 

26.725 

9 

UUULLLLLU 

0.444E 0 

0.341E-3 

31.142 

10 

UUUULLLLLU 

0.449E 0 

0.192E-3 

33.681 




0.189E-3 

33.758 

11 

ULLUUUULLLU 

0.291E0 

0.821E-4 

35.504 

12 

UUULUULLLLLU 

0.201E0 

0.352E-4 

37.558 

13 

1 ULUUUUULLLLLU 

0.199E0 

0.153E-4 

41.148 


1121 






TABLE 3 

Optimal design of 1-D beam structure, to minimise energy transmission, 23<Q<6L 

Original Energy; Non-dimensional kinetic energy in bay N of the periodic structure. 

Final Energy: Non-dimensional kinetic energy in bay N of the optimised structure. 



6 

LLLLUU 

7 

LLLULUU 

8 

LLLLLUUU 

9 

ULLLLLLUU 


0.494E 0 


0.183E0 

0.175E0 


0.648E-2 


0.246E-2 

0.180E-2 


III » 


UUULLLLLUU 


0.105E0 


0.277E-3 


2.762 

3.916 


18.821 


18.715 

19.878 


21.868 


25.787 


11 

UUULLLLLLUU 

0.105E0 

0.776E-4 

31.313 

12 

UlTULLLLLLLUU 

0.166E0 

0.526E-4 

34.991 

13 

UUULLLLLLULUU 

0.973E-1 

0.282E-4 

35.379 

14 

UUUULLLLLLULUU 

0.581E-1 

0.122E-4 

36.778 



















TABLE4 

Optimal design ofl-D beam structure, to minimise "maximum” strain energy. 


Bay No.: Bay in which the optimal minimum “maximum” non-dimensional strain energy occurs 

Original Energy: Initial “maximum” non-dimensional bay strain energy of the periodic structure. 

Final Energy: Non-dimensional strain energy in bay N’ of the optimised structure. 


No of 

Optimal 

Original 

Final 

Bay 

Reduction 

Bays, N 

Pattern 

Energy 

Energy 

No., N’ 

(dB) 

Q.=50 


9 

ULLLUUUUL 

0.667E-1 

0.297E-1 

1 

3.514 




0.296E-1 

1 

3.528 

10 

ULLLUUUULU 

0.540E 0 






















12 

ULLLULUULULU 

0.404E 0 

0.297E-1 

1 

11.336 




0.296E-1 

1 

11.351 


40<Q<60 

9 

UUUUULULU 

0.486E 0 

0.710E-1 

1 

8.354 




0.449E-i 

1-2 

10.344 

10 

ULULULULLL 

0.606E 0 

0.643E-1 

1 

9.743 




0.45 IE-1 

1-2 

11.283 

11 

ULULULULLUU 

0.456E 0 

0.682E-1 

I 

8.252 




0.425E-1 

1-2 

10.306 

12 

UUUIJUUUXJUUUL 

0.332E 0 

0.550E-1 

2 

7.808 




0.412E-1 

1-2 

9.062 


23<n<61 


9 

LLLLLLLLL 

0.234E 0 

0.203E 0 

1 

0.617 




0.979E-1 

1 

3.784 

n 

LLLLLLLLLL 

0.200E 0 

0.178E0 

1 

0.506 

■I 



0.95 IE-1 

1-2 

3.228 

11 

UUUUUUUUULL 

0.198E0 

0.193E0 

1 

O.lll 




0.910E-1 

1-2 

3.376 

12 

UUUUUOUIJULUU 

0.314E0 

0.182E0 

I 

2.369 




0.803E-1 

1-2 

5.922 


1123 























Phase £ AUenualion 


Figure 1; A simply supported periodic 


Itlijsj 











kinelic.enorgy in bay 12, T, 













1128 


EFFECTS OF GEOMETRIC ASYMMETRY ON VIBRATIONAL 
POWER TRANSMISSION IN FRAMEWORKS 

J L Homer 

Department of Aeronautical and Automotive Engineering 
and Transport Studies, Loughborough University 
Loughborough, Leics , LEll 3TU, UK 


ABSTRACT 

Many sources, such as machines, are installed on supports, or frameworks, 
constmcted from beam-like members. It is desirable to be able to predict 
which wave types will be present at particular points in the support structure. 
By using the concept of vibrational power it is possible to compare the 
contributions from each wave type. Wave motion techniques are used to 
determine the expressions for vibrational power for each of the various wave 
types present. The results from the analysis show the amount of vibrational 
power carried by each wave type and the direction of propagation. 
Consideration is given to the effect on the vibrational power transmission of 
introducing misalignment of junctions in previously symmetric framework 
structures. By splitting a four beam junction in to, say, a pair of three beam 
junctions separated by a small distance, it is possible to establish the effects of 
separating the junctions on the various transmission paths. Unlike other 
techniques using vibrational power to analysis frameworks, the model keeps 
the contributions from each of the various wave types separate. This allows 
decisions to be made on the correct vibrational control techniques to be 
applied to the structure. 


INTRODUCTION 

When attempting to control vibration levels transmitted from a machine 
through the various connections to the structure upon which it is mounted, it is 
desirable to be able to identify and quantify the vibration paths in the stmcture. 
Often large machinery installations are installed on frameworks consisting of 
beam like members. These frameworks are then isolated from the main 
structure. Simple framework models are also used in the initial design stages 
of automotive body shell structures to determine dynamic responses. 

If the dominant transmission path in the framework is identified it is 
possible to reduce vibration levels by absorbing the mechanical energy along 
the propagation path in some convenient manner. By utilising the concept of 
vibrational power it is possible to quantitatively compare the wave type 
contributions to each transmission patL In order to predict vibrational power 
transmission in a framework, it is necessary to identify the wave amplitude 
reflection and transmission coefficients for each joint in the structure. Lee and 
Kolsky [1] investigated the effects of longitudinal wave impingement on a 
junction of arbitrary angle between two rods. Similarly Doyle and Kamle [2] 
examined the wave amplitudes resulting from a flexural wave impinging on 
the junction between two beams. By using the reflection and transmission 
coefficients for different joints, it is possible to predict the vibrational power 
associated with flexural and longitudinal waves in each section of the 


1129 


framework. Previous investigations [3,4] have considered the effects of bends 
and junctions in infinite beams. This work was extended to consider the finite 
members which constitute frameworks [5]. Unlike other techniques [6, 7] 
utilising energy techniques to analyse frame-works, the technique produces 
power distributions for each wave type present in the structure. By comparing 
the results for each wave type, it is possible to apply the correct methods of 
vibration control. 

The technique is used to investigate the effect of geometric asymmetry 
on the vibration transmission, due to steady state sinosodial excitation, in a 
framework structure similar to, say, those used in the automotive industry 
(figure 1). By splitting a four beam junction into a pair of three beam 
junctions separated by a known distance, it is possible to establish the effect of 
junction separation on the dominant transmission paths. The investigation 
presented is limited to one dimensional bending waves and compressive waves 
only propagating in the structure. To consider the addition of other wave 
types ie. torsional waves and bending waves in the other plane, the analysis 
presented here for the junctions should be extended as indicated by Gibbs and 
Tattersall [3]. 


TRANSMITTED POWER IN A UNIFORM BEAM 

For flexural wave motion, consider a section of a uniform beam carrying a 
propagating flexural wave. Two loads act on this beam element, the shear 
force and the bending moment. It is assumed that the flexural wave can be 
described by using Euler-Bemoulli beam theory, so that the displacement can 
be expressed as 

W(x,t) = Af sin (cot-kfx), 

the shear force acting on a section as 

S = Eia3W/ax3, 

and the bending moment on the section as 
B = El 02 W/ax2. 

Then the instantaneous rate of working X at the cross-section is given 
by the sum of two terms (negative sign merely due to sign convention). 

0t 0x0t 0X^ 0X^ 3x0t 

The time averaged power 
{P)f = (1/T) I X dt then is given by (P)f = Elkf coAf 

I ( 1 ) 

For longitudinal wave motion consider a section of a uniform beam 
with a longitudinal wave propagating through the beam 

U (x, t) = Ai sin ( 0 )t-kix) 


1130 


The instantaneous rate of working X is then 
X= -EA(au/ax)u 


and the time averaged power is 



Xdt = rEA0)kiA| 


( 2 ) 


If dissipation is present in the structure, the modulus of elasticity may 
be considered to be a complex quantity 

= E(l + iTi) 


where represents the loss factor of the material, present due to 
inherent material damping. 

The displacement of a beam at a distance x from the source, due to 
flexural wave motion may now be considered to be, assuming that 
material damping is small. 


1 ^ 
-kri- 


W = Afe 4 

and the resulting time averaged power is given by 
(P}f = EIcok^e-'‘'’i2 Af 


(3) 


The above reduces to equation (1) at the source. 

Similarly, the displacement of beam, due to longitudinal wave motion 
may be considered to be 

T ^ 

-kiTi- 

U = Aie ^ sin(cot-kix) 

and the resulting time averaged longitudinal power may be rewritten as 


(P),=iEAcok,e-‘''^’‘A? 


(4) 


WAVE TRANSMISSION THROUGH A MULTI BRANCH JUNCTION 

Consider a four branch junction as shown in figure 2. Assuming only flexural 
and longitudinal waves propagating in the structure, the displacements of Arm 
1 will be, where A 4 represents the impinging flexural wave arriving from 
infinity. 


Wi(x,t) = (Aie^f’"" + A 3 e‘^fi'' + A 4 e e‘“^ 


(5) 


1131 


( 6 ) 


U,(x,t) = (Aae''‘‘i’‘)e‘“‘ 


Similarly for arms 2 to 4 the displacement will be, 

where ^ cos 0 n and n is the beam number 

W„(v„,t) = )e‘“‘ 

(V) 

( 8 ) 

Here A 3 , A 4 , 64 ^ are travelling flexural wave amplitudes', Ai and 
B 2 n are near field wave amplitudes and Aa and are travelling 
longitudinal wave amplitudes. 

In previous work [2] in this field a theoretical model was used in which 
it was assumed that the junction between the beams was a rigid mass. The 
mass or joint is modelled here as a section of a cylinder. This represents the 
physical shape of most joints in practical systems. It has been shown [4] that 
the joint mass has an insignificant effect on the reflected and transmitted 
power for the range of values used in this work. 

The joint mass Mj = pjTtL^J^/ 4, and the moment of inertia of the joint 
isIj=ML2/8. 

By considering the conditions for continuity and equilibrium at the 
beam junction the following expressions may be written. 

For each arm 

For continuity of longitudinal displacement 

L 3W 

Ui=Un COsen-WnSinen + -^^sinen 

For continuity of flexural displacement 

W,=u„sinen+W„cos0„-| |^(l + cose„) 

For continuity of slope 

8W; ^ awn 


1132 



For the junction 
Equilibrium of forces 


' ' 2 ' ' 3x2 J 


-vfp fax L32 w„ 

11 I aVS 2 3 v|;^ 


El Ai ^ + Mj ^ = i[E„ A„^cose„ +E„ I„^^sine„ 
3x J dt^ il^ 3V„ 3< 


E T a^W, 3 r„, L3W| 

El Ii —t^ + M; —T Wi- ——— 


-^ -I- iVi; —y 1 “ “ ^T" 

3x2 2 3x 


n ;^TI ^ W 

= Z En Ajj- ”Sin6n-EnIn 2 *^ COS0n 

1 I 5¥n 


WAVE MOTION AT A FORCED OR FREE END 

As indicated in figure 1, the framework has one forced end and one free end. 
Assuming the structure is only excited by a transverse harmonic force, the 
boundary conditions are as follows: 

at the forced end 


E« = Pe^«t 

3x2 


EI^ = 0 

3x2 


EA^ = 0 


Similarly at the free end the above boundary conditions apply with the 
exception that 

EI^ = 0 
3x2 


1133 



POWER TRANSMISSION THROUGH A FRAMEWORK 


The structure shown in figure 1 consits of one four-beam junction, two three- 
beam junctions and four two-beam junctions. From the equations detailed in 
the above two sections, it is possible to construct matrices of continuity and 
equilibrium equations for sub structures. These may be combined to obtain 
the overall matrix for the system. For the framework shown in figure 1, the 
size of the overall matrix is 60x60. This matrix may be solved to obtain the 
sixty unknown wave amplitude coefficients from which time averaged 
transmitted power for each beam may be calculated using equations (3) and 
(4). 

Normalised nett vibrational power is then calculated at the centre of 
each beam constituting the structure. Nett vibrational power may be 
considered to be the difference between power flowing in the positive 
direction and power flowing in the negative direction for each wave type. 
Normalised nett power is considered to be nett power divided by total input 
power. The input power to a structure may be calculated from the following 
expression [8] 

Input Power = ^ IFIIVI cos0 


where 0 is the phase angle between the applied force and the velocity 
of the structure at the forcing position. 


Figures 3-6 show the nett normalised power in each arm of a 
framework structure over the frequency range 0-lkHz excited by IN force, 
whose material and geometric properties are given in Appendix 2. For the 


results shown, angle 1 is 45® and angle x is 40® (or the ratio 



= 0.89) and 


L = y = 0.1m. Using these parameters the ratio of the length of beam No.6 to 
beam No.4 is 0.12. The predicted flexural power is shown in figures 3 and 4 
and from these it can be seen that the dominant transmission paths are arms 1 
and 5, the forced and free arms. The transmitted power in arm 10 is next 
dominant and comparable to arm 5 in the region 0-600Hz. 

The response for ail other arms are small, typically less than 5% of 
input power, with, as would be expected, arms 2 and 9 being approximately 
identical in transmission properties. 

Figures 5 and 6 show the nett normalised power for the longitudinal 
waves in the structure. As the frequency range of interest corresponds to a 
flexural Helmoltz number of 1 to 5 with L being the reference length, the 
conversion of power from flexural to longitudinal waves is minimal. From the 
figures it can be seen that beams 1, 5, 6 and 10 have identical transmission 
characteristics, which would be expected at such large longitudinal 
wavelengths. Significant longitudinal power is only observed in arms 3 and 8 
in the frequency region 200-300Hz. This frequency region coincides with a 
drop in the flexural power due to the structure being at resonance in that 
region. It should be noted that power transmitted through arms 3 and 8 has 
travelled through two junctions. 


1134 


EFFECT OF GEOMETRIC ASYMMETRY 


By altering the ratio of angle 6i to angle 0x it is possible to alter the length of 
beam 6 and hence move a pair of three arm junctions further or closer apart. 
From the discussion in the previous section, it was seen, for the structure 
under investigation, that the dominant flexural path, not surprisingly, is 
through the centre of the structure, whilst the peaks in longitudinal power 

occur in beams 3 and 8. Thus 0x was varied and the effect on transmission in 
the dominant paths noted. 

Figures 7-9 show flexural power for arms 5 and 10 and longitudinal 
power for arm 8 for four values of 0x. The values chosen were 36°, 38.25°, 
40° and 42.75° which are equivalent respectively to 0x over 0] ratios of 80%, 

85%, 90% and 95%. Thus as 0x increases, the structure moves to being 
symmetrical in nature. From figure 7, the increase in junction separation 
decreases power in the frequency region 0-500Hz and increases it in the region 
500-lkHz. In beam 10 (figure 10) the effect on the flexural power is reversed 
with increase in junction separation leading to increased power below 500Hz 
and decreased power above 500Hz. It should also be noted that increased 
junction separation has little effect on the power below 250Hz. The effect was 
also noted on all other beams which had both ends connected to a joint. It 
may be concluded that at long flexural wavelengths the junction separation has 
little effect with the impedance mis-match at the junctions being the important 
criteria to effect transmission. It should also be noted that increasing power in 
one arm ie. 5, causes a decrease in arms ie. 10, connected to it. An example of 
the effect of junction separation on longitudinal power is shown in figure 9. 
This shows nett normalised longitudinal power for arm 8 for the same 

variation in 0x. Again minimal effect is seen at low frequencies, with 
increased junction separation having different effects in different frequency 
region. Increasing junction separation has little effect on the region between 
200 and 300Hz when the longitudinal power was dominant. This would be 
caused by the junction separation having little effect on the structures flexural 
natural frequencies. Only by shifting those would the peaks in longitudinal 
power by shifted in frequency. 


CONCLUSIONS 

Results are presented for normalised nett time average vibrational power for a 
framework structure. The geometric symmetry of the structure is broken by 
allowing one angle to decrease in value. The effects of varying the angle 
change by up to 20% of its original value are investigated. Although the 
results presented are for one example only, highlighted are the fact that 
decreases in power in one part of the structure result in increases in power in 
another part. Also shown was the effect of splitting a junction in to a pair of 
junctions is minimal at low frequencies, or long wavelengths. From the results 
of the analysis it is possible to establish frequencies and positions for 
minimum power on the structure. Other configurations of framework 
structure may be analysed by applying the equations presented. 


1135 


REFERENCES 


1. J. P. LEE. and H. KOLSKY 1972 Journal of Applied Mechanics 39, 809- 
813. The generation of stress pulses at the junction of two non-collinear 
rods. 

2. J. F. DOYLE and S. KAMLE 1987 Journal of Applied Mechanics 54, 
136-140. An experimental study of the reflection and transmission of 
flexural waves at an arbitrary T-Joint. 

3. B. M. GIBBS and J. D. TATTERS ALL 1987 Journal of Vibration, 
Acoustics, Stress and Reliability in Design, 109, 348-355. Vibrational 
energy transmission and mode conversion at a corner junction of square 
section rods. 

4. J. L. HORNER and R. G. WHITE 1991 Journal of Sound and Vibration 
147, 87-103. Prediction of vibrational power transmission through bends 
and joints in beam-like structures. 

5. J. L. HORNER 1994 Proceedings of the 5th International Conference on 
Recent Advances in Structural Dynamics, SOUTHAMPTON UK, 450- 
459. Analysis of vibrational power transmission in framework structures. 

6. P. E. CHO and R. J. BERNHARD 1993 Proceedings of the 4th 
International Congress on Intensity Techniques, SENLIS, France, 347- 
354. A simple method for predicting energy flow distributions in frame 
structures. 

7. M. BESHARA and A. J. KEANE 1996 Proceedings of Inter-Noise '96, 
LIVERPOOL, UK 2957-2962. Energy flows in beam networks with 
complient joints. 

8. R.J.PINNINGTON and R.G.WHITE 1981 Journal of Sound and Vibration 
75, 179-197. Power flow through machine isolators to resonant and non- 
resonant beams. 


APPENDIX 1 - NOTATION 


A 

- Cross sectional area 

Q 

- Axial force 

Af 

- Amplitude of flexural wave 

s 

- Shear force 

Ai 

- Amplitude of longitudinal wave 

T 

- Time period 

B 

- Bending moment 

t 

- Time 

E 

- Young’s modulus 

U 

- Displacement due to 

E* 

- Complex Young’s modulus 


longitudinal wave motion 

F 

- Excitation force 

V 

- Velocity 

I 

- Moment of inertia 

w 

- Displacement due to 
flexural wave motion 

Ij 

Jw 

- Moment of inertia of joint 

X 

- Instantaneous rate of 
working 

- Joint width 

X 

- Distance 

- Loss factor 

kf 

- Flexural wave number 

0n 

- Angle of Arm n 

ki 

- Longitudinal wave number 

Pj 

- Joint density 

L 

- Joint length 

M 

- Moment force 

Mj 

n 

P 

- Joint mass 

<}> 

Phase angle 

- Beam number 

- Transverse force 

¥n 

- Distance along Arm n 


1136 


<P>f - Time averaged flexural power co - Frequency (rad/s) 
<P >1 - Time averaged longitudinal power 

APPENDIX 2 - MODEL PROPERTIES 

33mm 
6mm 
5GN/m2 
1180kg/m3 
0.001 


1 



Beam Breadth 
Beam Depth 
Youngs Modulus 
Density 
Loss factor 


Figure 1: Framework Structure 



iNett Normalised Power 









Nett Normalised Power 



Figure 5: Longitudinal Power - Beams 1-5 


(Beam 1-, Beam 2., Beam 3 . . Beam 4-, Beam 5 



Figure 6: Longitudinal Power - Beams 6-10 

(Beam 6-, Beam 7., Beam 8 . . Beam 9-, Beam 10 


1139 
























THE INFLUENCE OF THE DISSff ATION LAYER ON ENERGY 
FLOW IN PLATE CONNECTIONS 


Marek Iwaniec, Ryszard Panuszka 

Technical University of Mining and Metallurgy, 
Structural Acoustics and Intelligent Materials Group 
30-059 Cracow, al. Mickiewicza 30, Poland 


1. Introduction 


Dynamic behaviour of mechanical strucmres may be modelled on the basis of 
and with the help of mathematical apparams used in Statistical Energy Analysis (SEA) 
[5] The method is especially useful to calculate the statistical approach vibroacoustical 
energy flow in middle and high frequency range. With the help of a few parameters, 
such'as- modal density, damping loss factor, coupling loss factor and the value of 
input power, building linear equations set it is possible to describe the flow of 
vibroacoustical energy in a complicated stmcture. There is also a possibility of quick 
estimation of the influence of constmction method on the vibroacoustical parameters 
of the whole set. In the following work an exemplary application of one of the most 
frequently used software for calculating the flow of acoustic energy has been 
presented- AutoSEA programme [1]. The aim of the work is practical modelling of 
vibroacoustical energy flows through screw-connection of two plates and comparing 
quantity results with experimental (outcome) measurements. Equivalent coupling loss 
factor has been calculated for a group of mumally combined elements constimting a 
construction fragment. A comparison between the measured results and the value of 
coupling loss factor in linear joint (e.g. in welded one) has also been made. Using the 
method of fmite elements, the influence of rubber separator thickness on the value of 
the first several frequencies of free vibrations has been computed as well. 

2. Physical model of plate connection 

A connection of two perpendicular plates has been chosen for modelling the 
flow of vibroacoustical energy in mechanical joints. Connection diagram is presented 
in Figure 1. On the length of common edge the plates has been joined with anglesteel 
by screws. A rubber separator (4) has been placed between the excited plate and the 
anglesteel leg (3). 


1143 



Modelled stmcture 


3. SEA model 

In order to carry out the vibroacoustical analysis of the system using Statistical 
Energy Analysis a model of the examined strucmre has been built. It has been 
assumed that in every element of the construction only flexural waves propagate. 
Every plate and the rubber layer have been modelled with just one appropriately 
chosen subsystem. The anglesteel, however, has been modelled as a continuous 
connection of two plates having the dimensions which correspond to the anglesteel 
legs the plates themself being set at the right angle. 



Fig. 2. SEA model of a system Fig- 3. Modal densities of the subsystems 


1144 
















Using the SEA method we are able to describe the flow of the vibroacousdcal 
energy in middle and high frequencies with an algebraic equation set. The exMined 
system consists of five simple subsystems, of which only one is exited to vibration 
with applied force. The flow of vibroacoustical energy m the model presented is 
depicted with the following equation: 


^ 1 tot 

-^21 

0 

0 

12 

^ 2 tot 

32 

0 

0 

“'•123 

^ 3 tot 

-‘n43 

0 

0 

“''134 

h 4 tot 

0 

0 

0 

"'’145 


0 


tot 


w, 

0 


^2 tot 


0) 





0 

0 

• 

^3 tot 

= 

0 

-^54 


^4 tot 


0 

^ 5 tot 


^5 tot 


0 


^ ^ 4 - is a total coefficient of energy loss for every subsystem, 

rj’. '"' - intemarioss factor of the subsystem, 

ri.. - coupling loss factor between subsystems, 

E- - the mean vibrational energy in Af frequency band in i-subsystem, 
W - the input power carried into i-subsystem from outside. 


To determine the elements of the coefficients matrix in eq. 1 it is necessary to 
know [1,4,6] coupling loss factors (CLF) between structural subsystems and damping 

In the SEA model in question transmission of the acoustic energy occurs in two 
tvpes of connections between; 

the plate and the beam (the point joint of the beam which is parallel to the edge 

of the plate, (transfers flexural waves), . . „ 

two plates (linear connections and point joints transferring flexural waves). 


The coupling loss factor between the plate and the beam which vibrate in the 
flexural way (in the case of the point joint), is defined with following equation [1]. 


1.75c^ 




( 2 ) 


where: 

c - is the speed of flexural wave, 

T - transmission factor, 

Q - the number of point connections, 
Gj - angular frequency, 

A - the surface of the plate. 


1145 


The speed of the flexural wave in the first plate c„ can be calculated in the 
following way: 







(3) 


In the model under examination formula (2) defines coupling loss factors between the 
anglesteel leg (2) and the beam (4) - (factors 1 , 3 . and tiJ or the beam (4) and the plate 
(5) - rj 45 and 7 / 54 . 

The flow of energy between two plates (which are connected at the right angle 
and which vibrate in flexural way) is defined with the following formula in the case 
of linear connection: 


where: 

1 - is the length of the connection. 

With above formula it is possible to describe the flow of energy through correctly 
made welded joints of plates or, for example, through bent plates. In the system 
presented the factor determines the flow of energy between anglesteel legs ( 2 ) and 

(3) 

The coupling loss factor between two plates with a point joint is described by 
the following formula: 




iiL^ 

3 (o.X, 




(5) 


This type of connection occurs between the plate representing the anglesteel 

leg and the plate (5). ^ . 

After defining the value of factors matrix in the first equation it is possible to 

specify the ratio of the vibroacoustical energy gathered in plate (1) and (5). 

_ (n 2tof^ 3t0t~ ^ 24^ 32 ) 4fot^ 5tnt~ ^ 54^ 45^ ~ 34^ 43^ 5tot (5) 

£5 ^ 21 ^ 32 '^ 43^ 54 

The damping loss factor is important parameter of every subsystem. For steel 
plates used in the experiment the value of the damping loss factor have been measured 
experimentally with the decay method. The results of the measurements have been 
presented in figure 4. The frequency characteristic of rubber damping has been shown 
in figure 5. 


1146 



DLF of steel [-] 



frequency [Hz] frequency [Hz] 


Fig. 4. Damping loss factor of steel Fig. 5. Damping loss factor of rubber 
4. Experimental research 

Experimental investigations have been carried out for connections made with 
the use of rubber separator (elastic layer) of 50° Shore hardness. The connection was 
build up of two identical, perpendicular plates connected each to another using the 
anglesteel and the elastic rubber layer. These are the properties and material 
parameters of individual elements: 
plates: 

- material constructional steel (St3); 

- dimensions: 500 * 500 * 2.2 mm; 

- Young modulus: 2,1 10'^ Pa. 
anglesteel: 

- material: constructional steel (St3); 

- dimensions: L 40 40 2.2 mm; 

- Young modulus: 2,1 10'‘ Pa 
elastic layer: 

- material: rubber 50° Shore 

- dimensions: 500 40 mm 

- thickness: 2, 3, 4, 5, 6 mm 

To avoid the loss of mechanical energy in the environment, during measure¬ 
ments the construction was suspended to the supporting frame with three weightless 
strings in such way that only rigid body motions in the plane perpendicular to the plate 
surface can occur. The excitation of the wide-band type with constant power spectral 
density was applied in the symmetry axis of the plate (5) about 20 mm below the 
upper edge (Figure 1). During the experiment the distribution of vibrating velocities 
on the plate surface was obtain by non contact method using laser-vibrometry. 


1147 







5. FEM model 

Vibration of modelled structure in low frequency has been analyzed by toe 
Finite Element Method (FEM). Several FEM models, was build in order to consider 
L valid thickness of the mbber layer. The mbber thickness has been from 

0 mm to 6 mm. The dimensions and material parameters of the plates and the 
LSesteel was constant. The structure was fixed in four comers. Calc^auons were 
mfde for a division of the strucmre into 608 elements of type BRI^8. The mfluence 
of the rubber thickness on the eigenfrequencies was remarkable. The results for first 
15 eigenfrequencies are presented in the table 1. In fig. was shown also the changes 
of the value of natural frequencies in comparison with the natural frequencies of 
strucmre without applying the mbber layer. 


table 1. Namral frequencies of connection 


LaDlc i. i>aLutai ai. — - 

1 Natural frequencies (Hz) for various rubb 

■ ■. - - • 

er layer tlrickness 


0 nun 

2 nun 

3 mni 

4 nun 

5 nun 

127.8 

123.1 

121.5 

119.4 

117.5 

297.5 

235.8 

224.6 

212.6 

204.6 

507.6 

432.5 

391.3 

357.1 

330.0 

1003.2 

566.9 

525.7 

503.7 

491.9 

1282.3 

807.1 

768.9 

733.5 

712.2 

1900.5 

1185.2 

1169.4 

1123.2 

1070.2 

2386.7 

1570.2 

1483.6 

1376.2 

1304.6 

3272.3 

1851.6 

1634.2 

1531.6 

1490.8 

3603.4 

2303.1 

2263.4 

2167.8 

2147.3 

4263.3 

2611.2 

2539.5 

2496.9 

2463.7 

4759.5 

3192.7 

3515 

3110.1 

2913.2 

5438.4 

3926.7 

3874.6 

3626.4 

2937.0 

7627.4 

4110.1 

4105.5 

3639.9 

3026.6 

7691.6 

5070.7 

5137.0 

3670.5 

3111.2 

11543.2 

5547.1 

5503.9 

3721.9 

3181.2 


The decreasing of the absolute values of the natural frequences is observed according 
to increasing of the thickness of the rubber layer. Beginning from die third of 

°ib adons of the system the decrease of the natural frequencies is almost constant for the 
"layer 2 mm or 3 mm and is continuously decreasing for rubber layer 4-6 mm (See 

fig. 6). 


1148 








Fig. 6. Changes of natural frequencies 

6. Comparison with experimental results 

The equivalent coupling loss factor defining the energy flow between the plates has 
been determined experimentally [31 for a model consisting of two subsystems; two plates. 
The coupling loss factor in such two element model may be specified by the following 

equation: 


1 zast. 


^2 ^1101 

N, E2.0C 


- 1 


(7) 


The quotient of plate energies E,/E, in a two-element model is relevant to the quotient of 
energy in the first and fifth subsystem (E./E,) in the five element model presented m figure 
0 the value of these quotient is defined with the equation (6) 

In the picture we have presented the values of the equivalent coupling loss factor in 
the connection. Individual points in the diagram show the results obtained experimentally. 
The values received in computer simulation have been presented as a continuous diagram. 
The upper curve shows the values of the coupling loss factor in the joint before the 
application of the rubber layer. 

In the frequency range above ca. 125 Hz we have received a very good comparison 
of experimental results and computer simulation results performed with the AutoSEA 
software At the frequency of about 200Hz there occurs a local minimum of the equivalent 
coupling loss factor between the plates. The value of the minimum is essentially influenced 
by the value of the rubber damping loss factor. The frequency (with the minimum CLF) is 
strongly influenced by the peak frequency of the rubber damptng curve. 


1149 










o.ooo2ii- 


symulation 


experiment 


welded plates 


16' ‘ '40' ' ’lOO 250 630 1600 4000 

frequency [Hz] 

Fig.7 Equivalent CLF factor in the joint 

7. Conclusion 

A way of modelling the vibroacoustical energy flow with the help of SEA method has 
been presented in the work. We have examined the screw connection of two plates, where 
a rubber elastic layer has been applied, A comparison has also been made between the results 
of computer simulation of the mechanical energy flow with SEA method and the experimental 
results, and thus we have noticed the good correlation, especially as far as middle frequencies 

The joint modification through introduction of the rubber separator has a remarkable 

impact on the acoustic energy flow: , ^ 

- The application of the elastic layer in the Joint in question lowers the value ot 

eauivalent coupling loss factor in the whole frequency range. 

- the value of rubber damping factor has most significant influence on the acoustic 

eneroy flow through connection in middle frequencies range: 

“ The minimum value of the equivalent coupling loss factor in a joint is essentially 
influenced by the rubber separator damping loss factor. 

* The frequency of minimum CLF occurrence is strongly influenced by the peak 
frequency of the rubber damping curve. 

The increasing thickness of the rubber layer produces on decreasing natural frequencies of 
the structure. 


8. Bibliography: 

III AutoSEA - User Guide Vibro-Acoustic Sciences Limited 1992. 

121 Fahy F.J. Sound and Structural Vibration; Radiation, Transmission and Response Academic Pres 
nTkirtuin J Smals N. Panuszka R. "Method of estimating the coupling loss factor for a set of 

nlates" Mechanic, Technical University of Mining and Metallurgy, Cracow, 10,1991. 

[^41 Lalor, N.: The evaluation of SEA Coupling Loss Factors. Proc. V School Energy Methods in 

Vibroacoustics" - Supplement, Krakow-Zakopane 1996. , . „ „ 

[51 Lyon R., DeJong R.; Theory and Application of Statistical Energy Analysis. Butterworth- 

Heinemann, Boston, 1995. 


1150 







Variation Analysis on Coupling Loss Factor 
due to the Third Coupled Subsystem in 
Statistical Energy Analysis 

Hongbing Du Fook Fah Yap 
School of Mechanical & Production Engineering 
Nanyang Technological University 
Singapore 639798 


Abstract 

Statistical Energy Analysis (SEA) is potentially a powerful method for 
analyzing vibration problems of complex systems, especially at high frequen¬ 
cies. An impoitant parameter in SEA modeling is the coupling loss factor 
which is usually obtained analytically based on a system with only two cou¬ 
pled elements. Whether the coupling loss factor obtained in the classical way 
is applicable to a practical problem, which normally comprises of more than 
two elements, is of importance to the success of SEA. In this paper, the varia¬ 
tion of coupling loss factor between two subsystems due to the presence of a 
third coupled subsystem is investigated. It is shown that the degree to which 
the coupling loss factor is affected depends on how strong the third subsystem 
is coupled. It also depends on the distribution of the modes in the coupled sub¬ 
systems. This kind of effect will diminish when the damping is high, subsys¬ 
tems are reverberant, or ensemble-average is considered, but not for individual 
cases. 


1 Introduction 

SEA is potentially a powerful method for analyzing vibration and acoustic problems 
of complex systems, especially at high frequencies, because of the simplicity of 
its equations compared to other deterministic analysis techniques. SEA models a 
system in terms of interconnected subsystems. The coupling parameter between 
any two subsystems is characterized by a coupling loss factor. If the coupling loss 
factors and internal (damping) loss factors of all subsystems are known, the power 
balance equation (e.g., see [1]) for each subsystem can be established. From this set 
of equations, SEA predicts the system response (due to certain types of excitation) 
in terms of the average energy of every subsystem. The energy can in turn be related 
to other response quantities such as mean velocity or strain. 


1151 


Historically, the SEA power balance equations were initially derived from an 
analysis of two coupled oscillators [2,3]. It has been shown that the energy flow 
between them is directly proportional to the difference in their uncoupled modal 
energies. The theory has then been extended to systems with multi-coupled sub¬ 
systems (e.g., [4]). Strictly this extension is only applicable if certain assumptions 
are justified [4,5]. Also the new concept of indirect coupling loss factor, which 
is used to represent the energy flow proportionality between the indirectly coupled 
subsystems, is also introduced. 

In practice, the indirect coupling loss factors are normally ignored in SEA ap¬ 
plications because they are very difficult to determine analytically. Only coupling 
loss factors between directly coupled substructures are considered. Some analy¬ 
ses [6-8] have shown that this approximation may lead to significant errors in the 
predicted results if certain conditions are not met in the system. These conditions 
include not only the well known requirement of weak coupling between subsystems 
(e.g., see [6]), but also others, such as given by Langley that the response in each 
element must be reverberant [7]; and by Kean that there should be no dominant 
modes (peaks) inside the frequency-averaging band [8]. However, it is usually dif¬ 
ficult to know whether these conditions are satisfied for a particular system. In fact 
the above mentioned conditions do not always hold for practical engineering cases. 
On this point of view, the importance of a coupling loss factor for describing the 
coupling between indirectly coupled subsystems are to be further examined. 

A related question is whether the coupling loss factors obtained from the system 
with only two subsystems can still be applied when other subsystems are present. 
Generally, the coupling loss factor is sensitive to the amount of overlap between 
the modes of the two coupled subsystems. When additional subsystems are cou¬ 
pled to the original two-subsystem model, the mode distributions of the originally 
coupled two subsystems will be affected. The change of mode distributions will fur¬ 
ther affect the modal overlap between the coupled two subsystems and finally the 
coupling loss factor between them. However, general estimation methods for cou¬ 
pling loss factor assume that the coupling parameters between two subsystems are 
not affected much by the presence of the other subsystems. Therefore the conven¬ 
tional approaches of deriving coupling loss factor are mostly based on consideration 
of a two-subsystem model only. One method is the wave approach, by which the 
coupling loss factor used in the SEA applications are derived analytically from aver¬ 
aged transmission factors of waves that are transmitted through a Junction between 
semi-infinite subsystems. This method only takes into account local properties at 
the joints and sometimes may be inaccurate. Recent research [1,7,9-11] based 
on the model with two-coupled subsystems has shown that the coupling parameter 
does depend on other system properties, such as damping loss factor, etc. It can be 
argued that, if there is a third coupled subsystem, the coupling parameters between 
the first two subsystems will also depend on the energy flow to the third subsys¬ 
tem. Therefore, from a practical point of view, the coupling loss factor estimated 
for two-coupled subsystems, ignoring the indirectly coupled subsystems, can only 
be of approximate value. 


1152 


In this paper, the variation of coupling loss factor between two subsystems due 
to the presence of a third coupled subsystem is studied. In the following sections, 
the coupling loss factor is firstly expressed in terms of global mobility functions. 
The exact solution of mobility functions is only for simple structures. However, for 
general structures, it can be obtained by Finite Element Analysis (FEA) [10,11]. 
The coupling loss factors obtained respectively in the cases with and without the 
third subsystem in the model are compared for two particular system configura¬ 
tions, respectively. The system used in this investigation is one-dimensional simply 
supported beanis coupled in series by rotational springs. By varying the spring stiff¬ 
ness, the strength of the coupling between the second and the third subsystems can 
be changed. It is shown that the effect of the third coupled subsystem on the cou¬ 
pling loss factor between the first two coupled subsystems depends on how strong 
the third subsystem is coupled. For each individual case, it is also shown that this 
kind of effect may be positive or negative, depending on the distribution of modes 
in the coupled subsystems. 


2 Coupling Loss Factor by Global Modal Approach 


In this section, a modal method is used to derive coupling loss factor in a sys¬ 
tem with any number of coupled subsystems. The result is then simplified for two 
cases: (1) a three-subsystem model; (2) a two-subsystem model which is simply 
substructured from the previous three-subsystem model by disconnected the third 
subsystem. 

For a linear system which consists of N coupled subsystems, if “rain-on-the- 
roof” excitation [10] is assumed to be applied to each subsystem in turn, the corre¬ 
sponding response energy can be expressed as 

I I drdsdu, (1) 

sith.'ii/sl.Kmi Mii.bsy.'it.emj 

where /?/; is the total time-averaged response energy of the subsystem i. due to 
the excitation on the subsystem j, is the transfer mobility function be¬ 

tween the response points r and the excitation point H is the averaging range 
of frequency, u.', in and S are the mass density and the power spectral density of 
excitation. The input power due to the excitation is given by 

I RelH(s,.i,L 0 )]d.sdu (2) 

Q sahsyale'inj 


where is the real part of the point mobility at the position 

simplicity, two terms, a/, and are defined as 


a,, = 


i>, = ^ = 


Al 

m;S j 

Ik 


\ H[r, s,uj)\~ dr ds cl.in 


n sii.bsy.'iUjvii /iubfiyslenij 

lle[H{.^,s,ij)] dsd.u: 

n subsyslr;mj 


For 


(3) 

(4) 


1153 


The mobility function, H{r,.s,uj), is to be expressed in terms of the global modes 
of the system, which can be obtained by Finite Element Analysis (FEA). By the 
principle of reciprocity of the mobility function, the relation of aij = a ji always 
holds regardless of the strength of coupling and the magnitude of input power if the 
excitation is “rain-on-the-roofTheoretically applying the Power Injection Method 
[12] we can obtain the SEA equation as 

n = [77]E (5) 

where H = {Hi, ila, - • • , and E = {E[, E-z, - - - , E^r}'^. The SEA loss 
factor matrix [ 77 ] is 


iVl + 


i.^2 


(7?yV + 


■niiau 


niiCLiisr 

bi 

b-z 

bN 

■nua-ii 


■nizCL-zN 

bi 

bo 

bjV 

77?.;Va,Yi 


•m.A/a/VY 

^1 

Ih 

b^! 


= — B A“'M- 

UJr 


where, //; is the internal loss factor for subsystem i., v/,, is the coupling loss factor 
from subsystem -I. to subsystem j, ujc is the central frequency of the averaging band 

n. 


A = [a,,] 


From equation ( 6 ), the reciprocity principle of the coupling loss factors can be easily 
seen, due to aij = aji. 




1 

''■bi 

, M = 




L ’-J 


Vij n?.; b j ni jbj/ (cu^/i / 2) 

i]ji niibi ■niibi/iujc'^l'l) 


Where, ( 77 ?.; 6 ,:)/(u,v 7 r/ 2 ) is the averaged real part of point mobility [4] and can be 
regarded as the generalized modal density of the subsystem i. Assuming weak 
coupling and light damping, it approximately equals to the classical definition of 
modal density [13]. Therefore, the relation given by equation (7) also reduces to the 
classical reciprocity principle. 


1154 


2.1 Substructured two-subsystem model 



Figure 1: A general SEA model with three coupled subsystems in series 


Consider a whole system with three substructures coupled in series as shown in 
Figure 1. If subsystem 3 is removed, the coupling loss factors between subsystems 
1 and 2 are given in the equation 

biCL22 bi(Li-2 

"h'^712 — ^?2L _ ^ n?. i(fi 1 [fi 22 — '^7.2(^11^22 — <'^i 2 ^^' 2 l) 

— 7/12 772 + 7/21 u-V _ ^20.21 _ _ ^2(111 _ 

777 i{an<^f 22 “ <^^ 12 f'' 2 l) 7772(^110-22 — O12O21) 

r bj ^to -12 1 (8) 

^ J_ 777 ifli| 7772011022 

62O21 ^2 

7771O11O22 777.2 0 22 

The approximation in the above equation is due to 011022 » 012 O 21 when the cou¬ 
pling is weak. Manipulating equation (8) with or without using the approximation 
both can work out the coupling loss factors 7/12 and 7/21 as 



(9) 


r 61 6-, 

(777 1 I ) (77 72 62 ) - 7/2 -77 

07^-777-101 1 07^777.2022 

! , ^2 . f. *^1 

77 7 20'7- 777 [Oi 

"07c7772022 07^-777-1 011 


( 10 ) 


The equations are true regardless of the strength of the coupling. It can be seen that 
7/12 and 7/21 depend on the values of the three terms 777 .,+/, 7 // and bij{u:^ni-,au). The 
first two are the generalized modal density and the internal loss factor, or in combi¬ 
nation equivalent to modal overlap factor. The third one, by noting the definitions 
of 6; and an, is the ratio of input power to response energy for the directly excited 
subsystem, i.e., the total loss factor of subsystem i. From equation (8), this term 
can be approximately expressed as 


Total loss factor 
of subsystem i 


b; 

miLOcCLii 


m + 5I'+-.7 


( 11 ) 


1155 






In the classical wave approach, where semi-infinite subsystems are assumed, the 
total internal loss factors becomes 

■n total, I = m + Vn and l]iot.al,2 = 112 + vTi (^2) 

where is the classical coupling loss factor. Substituting equation (12) into equa¬ 
tions ( 9 ) and ( 10 ), i]ij reduces to the classical iiff, which only depends on the local 
properties at the joints rather than other properties of the system, such as damping. 
If the subsystem modal parameters are used to evaluate the term, bil{ujcmiaii), then 
the total loss factor is Just the internal loss factor of the subsystem and the cou¬ 
pling loss factor is equal to zero. This is reasonable because using the uncoupled 
modal parameters instead of the coupled modal parameters is actually equivalent to 
removing the coupling between two subsystems. 

However, for finite system where the assumption of semi-infinity is not justified, 
there will be no immediate simplication for equations (9) and (10). Numerically, 
FEA can be employed to obtain the global modes and then the coupling loss factor 
can be calculated [ 10 , 11 ]. 


2.2 Full three-subsystem model 

Instead of substructuring, consider the three-subsystem model as a whole system, 
shown in figure 1. Now the order of equation ( 6 ) is reduced to 3. With the global 
modal parameters obtained from FEA, the coupling loss factors can be directly 
evaluated. However, when the coupling between subsystems is weak, the order- 
reduced equation ( 6 ) is still able to be simplified. Matrix A may be alternatively 
expressed in the form of 



■ a, L 

0 

0 ‘ 


■ 0 

«L2 

0 


■ 0 

0 

«13 ' 

A - 

0 

(t-ll 

0 

-f 

a\2 

0 

d.23 

+ 

0 

0 

0 


0 

0 

«33 . 


0 

((•23 

0 . 


. (l.[3 

0 

0 


where the terms on the right side are sequentially defined as Ai, A 2 and A 3 . Under 
the assumption of weak coupling, the non-zero entries in Ai, A 2 and A 3 will be of 
the order O(t^), (9(e^) and respectively [14], The inverse of matrix A may 

be approximately written as 

= Ai”^ — Ai“^A2Ai”^ — Ai"^A3Ai ^ 4 -Ai ^A2Ai ^A2Ai 


+ ••• 


(14) 


As an approximation, substituting only the first three terms in equation (14) into 
equation ( 6 ) gives 


[tj] ^ l/uv- 


/>■ 

bo f t’12 
rn I f( 11 a 2 2 
b3(l-\3 

ni[a\ KM.3 


■ni2a 11 (1-22 

b2 

1112(1.22 

63^23 

1112 ( 1 . 22^(33 


bi(t.\3 
in3(l\\(l.33 
bid. 13 
in.3Ct.2'i((33 

63 ~ 

1((3((33 


(15) 


1156 



Generally, under the condition of weak coupling, the indirect coupling loss factors, 
■ihs and 7731 , are much smaller than the direct coupling loss factors and the internal 
loss factors [5,7,15]. The diagonal elements in [rj] can therefore be approximated to 
the sum of internal loss factor and direct loss factor. It can be shown from equation 
(15) that equation (11) remains valid for three coupled subsystems. But in the three- 
subsystem case, the term, b,|{uJcm.^au), is to be evaluated by using the global modes 
of the three-subsystem model. 


3 Numerical Examples and Variation Analysis 

In this section, two examples with different configurations are used to show the vari¬ 
ation of coupling loss factor due to the presence of a third coupled subsystem. The 
coupling loss factor of the two-subsystem model is evaluated by using equations (9) 
and (10). For the three-subsystem model, equations ( 6 ) and (15) are used. It can be 
shown that both equations ( 6 ) and (15) give the same results as the couplings are 
weak. 


3.1 Structural details and SEA model 


Ki 

beam I (TJT] - 

(a) ^ -Zi. 

beam 1 rnn beam 2 (TTH ^ 

(b) ^ ^ -A A- 


Figure 2; A structural model comprising of three beams 

To begin example calculations, consider initially a two-subsystem model (figure 2(a)) 
which is two thin beams coupled through a rotational spring. The group of flexural 
vibration modes of each beam are taken as a SEA subsystem. The spring provides 
weak coupling between them where only rotational moment is transmitted. When 
beam 3 is connected at the free end of beam 2 to the original two-beam model, 
a three-subsystem model is formed (figure 2(b)). The specifications for the three 
beams are given in table 1. The spring stiffness, A' 2 , is adjustable in order to look 
into the significance of the effect of the third subsystem. There are two cases where 
the length of beam 2 is: (i) L 2 - 1.0?72; (ii) L 2 = l-lm. The spring constants at 
the joints are chosen to be weak enough to ensure that:(a) the coupling loss factor 
is much smaller than the internal loss factor; (b) the indirect coupling loss factor is 
much smaller than the direct coupling loss factor. 

In the global modal approach (see section 2 ), the modes of two-subsystem model 
and three-subsystem model are obtained from FEA. In numerical simulation, the 


1157 


Table 1: The specifications of the three beams 


BEAM 

1 

2 

3 

length (m) 

2.0 

1.0 & 1.1 

0.7 

width (mm) 

4 

Thickness (mm) 

2 

Density (Kg/m^) 

7890 

Young’s Modulus (N/m'^) 

196E+9 

Poisson Ratio 

0.29 

spring constant, A*i (Nm/rad) 

1.0 


central frequency is 200Hz and the averaging band is selected as 100 ~ 300Hz. In 
order to take into account the contribution from the modes out of the band, all the 
modes up to 500Hz are extracted for evaluating the mobility functions in averaging. 
The modal loss factor is assumed to be the same for each modes used in averaging. 
This means that the internal loss factor is the same for each subsystem and is equal 
to the modal loss factor [11]. The results given are plotted against the modal loss 
factor in order to show the damping effect at the same time. 


3.2 Results and discussion 

Figure 3 shows the identified coupling loss factor 7/12 for the case (i) {Lo = 1.0777) 
with different stiffness of /v' 2 . The case of /v '2 = 0 means that the third subsystem 



Figure 3: 7/12 is negatively affected in three-subsystem model 
is not present. It can be seen that the coupling loss factor 7^12 is decreased in the low 


1158 









range of damping while the strength of the coupling between subsystem 2 and 3 is 
increased. The stronger the coupling, the more ijn is decreased. 

On the other hand, for the case (ii) where L 2 = l.lm, the different results are 
shown in figure 4 where the presence of the third subsystem would mainly increase 
i]i 2 in the low range of damping. The increasing magnitude is also dependent on the 
strength of coupling between subsystem 2 and 3. The explanation for the different 
variation trends of ?]i 2 due to the third coupled subsystem between figure 3 and 4 
will be given later. 



Figure 4; 771 ■; is positively affected in three-subsystem model 

From figure 3 and 4, the effect of damping on the coupling loss factor can also be 
observed. In the low damping region, increasing damping would increase coupling 
loss factor. After a certain turnover point, increasing damping would make the 
coupling loss factor decrease and finally 7712 becomes convergent to a value. This 
agrees with the conclusions drawn in [10, 11]. It is shown that, even though the 
length of beam 2 has a slight difference in figure 3 and 4, the converged values are 
still very close. Thus, the converged value seems not to depend on the variation of 
coupling strength at A '2 and the structural details, although, with the third subsystem 
existing in the system, the convergent speed is faster. Therefore, it is reasonable to 
believe that the converged coupling loss factor at sufficiently high damping only 
depends on the property of the joint rather than other system properties. This joint- 
dependent property of coupling loss factor in the high range of damping accords 
with the assumption in the wave approach. Here, it is convenient to define the 
convergent region in the figure 3 and 4 as the “joint-dependent zone”. 

However, before the “joint-dependent zone”, coupling loss factor seems very 
sensitive to the variation of damping loss factor as well as the strength of coupling 
between subsystem 2 and 3. It is because in the low damping region the system 


1159 






modal properties have been playing a major role in determining coupling loss fac¬ 
tor [10,11]. In general, the coupling loss factor represents the ability of energy 
transmitted between subsystems. It depends not only on the physical strength of 
the coupling (e.g., spring stiffness in the examples), but also on the amount of over¬ 
lap between the modes of two connected subsystems. The higher modal overlap 
between the modes of two connected subsystem, the more energy is transmitted 
between the subsystems. As a result, the coupling loss factor will be higher even 
though the physical strength at the joint is unchanged. If the modes in one subsys¬ 
tem are distributed exactly the same as those in one another(for instance, two exactly 
same structures are coupled together), the coupling loss factor would be varied to 
the maximum, and vice versa. Therefore, this region could be likely defined as 
“modal-sensitive zone”. 

In the “modal-sensitive-zone”, the dependence of coupling loss factor on the 
amount of overlap between the modes of two connected subsystems has been clearly 
shown in figures 3 and 4. For the case (i) illustrated in figure 3, the length of beam 2 
is half of beam 1 . Due to the characteristic of mode distribution in beam structure, 
the amount of overlap between the modes of subsystem 1 and 2 is more than that in 
the case (ii) shown in figure 4, where beam 1 is 2 meters and beam 2 is 1.1 meters. 
Therefore, case (i) has higher coupling loss factor than case (ii) in “modal-sensitive 
zone”. When the third beam is coupled, the induced variation of coupling loss factor 
depends on how the amount of overlap between the modes of subsystems 1 and 2 is 
affected. It can be increased or decreased and thus the coupling loss factor between 
subsystems 1 and 2 can also be increased or decreased due to the third coupled 
subsystem. For example, the amount of such overlap in case (i) is decreased after 
the third subsystem is coupled. As a result, the coupling loss factor, 7712 , becomes 
decreased. 

The above discussed variability of coupling loss factor due to the third coupled 
subsystem has been shown for individual cases. On the other hand, if an ensemble 
of similar structures are considered, this sensitivity may be reduced (as it is some¬ 
times positive or negative depending on each special situation). However, such a 
variability obtained from two typical examples is nevertheless very useful when 
one individual case is studied in SEA or SEA-like problems. The ignorance of such 
effect of the other coupled subsystems on the coupling loss factor may become one 
of the possible error sources causing SEA failure. 


4 Conclusions 

The variation of coupling loss factor due to the third coupled subsystem is stud¬ 
ied in this paper. The effect of a third coupled subsystem on the coupling loss 
factor between the first two coupled subsystems depends on how strong the third 
subsystem is coupled. Roughly, along with the damping in the subsystems, “joint- 
dependent zone” and “modal-sensitive zone” are defined according to the different 
variation properties of coupling loss factor. In the “modal-sensitive zone”, the ef- 


1160 


feet of a third coupled subsystem on the coupling loss factor could be positive or 
negative. It depends on how the amount of overlap between the modes of two con¬ 
nected subsystems is affected. This “modal-sensitive” effect may be averaged out 
for an ensemble of structures, but it is important when SEA is applied to individual 
cases. In the “joint-dependent zone”, the coupling loss factor is insensitive to the 
strength of the coupling between the second and third subsystems. Since the two 
different zones are allocated according to the system damping (which is equivalent 
to modal overlap factor when the central frequency and modal density are fixed), 
it shows the importance of reverberance in subsystems when the classical SEA is 
applied [7,8]. How to take into account the effect of the other coupled subsystems 
in evaluating coupling loss factor, especially when the system damping is low and 
when an individual case is considered, definitely needs to be further investigated. 


References 

[ 1 ] Richard H. Lyon and Richard G. DeJong. Theory and Application of Statistical 
Energy Analysis. Butterworth-Heinemann, second edition, 1995. 

[2] Richard H. Lyon and G. Maidanik. Power flow between linearly coupled os¬ 
cillators. Journal of the Acoustic Society of America, 34:623-639, 1962. 

[3] Eric E. Ungar. Statistical energy analysis of vibrating systems. Transactions 
of the ASME, Journal of Engineering for Industry, pages 626-632, November 
1967. 

[4] F. J. Fahy. Statistical energy analysis. In R. G. White and J. G. Walker, editors. 
Noise and Vibration, chapter 7, pages 165-186. Chichester, Ellis Horwood, 
1982. 

[5] J. M. Cuschieri and J. C. Sun. Use of statistical energy analysis for rotating 
machinery, part II; Coupling loss factors between indirectly coupled substruc¬ 
tures. Journal of Sound and Vibration, 170(2): 191-201, 1994. 

[6] P. W. Smith. Statistical models of coupled dynamical systems and the transi¬ 
tion from weak to strong coupling. Journal of the Acoustic Society of America, 
65:695-698, 1979. 

[7] S. Finnveden. Ensemble averaged vibration energy flows in a three-element 
structure. Journal of Sound and Vibration, 187(3);495-529, 1995. 

[8] A. J. Keane. A note on modal summations and averaging methods as ap¬ 
plied to statistical energy analysis (SEA). Journal of Sound and Vibration, 
164(1); 143-156, 1993. 

[9] B. R. Mace. The statistical energy analysis of two continuous one-dimensional 
subsystems. Journal of Sound and Vibration, 166(3):429-461, 1993. 


1161 


[ 10] Hongbing Du and Fook Fah Yap. A study of damping effects on coupling loss 
factor used in statistical energy analysis. In Proceedings of the Fourth Inter¬ 
national Congress on Sound and Vibration, pages 265-272, St. Petersburge, 
Russia, June 1996. 

[11] Fook Fah Yap and J. Woodhouse. Investigation of damping effects on statis¬ 
tical energy analysis of coupled structures. Journal of Sound and Vibration, 
197(3):35I-371, 1996. 

[12] D. A. Bies and S. Hamid. In situ determination of loss and coupling loss 
factors by the power injection method. Journal of Sound and Vibration, 
70(2): 187-204, 1980. 

[13] L. Cremer, M. Heckl, et al. Structure-Borne Sound: Structural Vibrations and 
Sound Radiation at Audio Frequencies. Springer-Verlag, second edition, 1987. 

[14] R. S. Langley. A derivation of the coupling loss factors used in statistical 
energy analysis. Journal of Sound and Vibration, 141(2):207-219, 1990. 

[15] J. C. Sun, C. Wang, et ai. Power flow between three series coupled oscillators. 
Journal of Sound and Vibration, 189(2) :215-229, 1996. 


1162 


THE EFFECT OF CURVATURE UPON VIBRATIONAL 
POWER TRANSMISSION IN BEAMS 
SJ. Walsh(l) and R.G.White( 2 ) 


( 1 ) Department of Aeronautical and Automotive 
Engineering and Transport Studies 
Loughborough University 

( 2 ) Department of Aeronautics and Astronautics 
University of Southampton 

ABSTRACT 

Previous research into structural vibration transmission paths 
has shown that it is possible to predict vibrational power 
transmission in simple beam and plate structures. However, in 
many practical structures transmission paths are composed of more 
complex curved elements; therefore, there is a need to extend 
vibrational power transmission analyses to this class of structure. 
In this paper, expressions are derived which describe the vibrational 
power transmission due to flexural, extensional and shear types of 
travelling wave in a curved beam which has a constant radius of 
curvature. By assuming sinusoidal wave motion, expressions are 
developed which relate the time-averaged power transmission to 
the travelling wave amplitudes. The results of numerical studies 
are presented which show the effect upon power transmission 
along a curved beam of: (i) the degree of curvature; and (ii) various 
simplifying assumptions made about the beam deformation. 

1 . INTRODUCTION 

Previous research into structural transmission paths has 
shown it is possible to predict vibrational power transmission in 
simple beam and plate structures. More recently, transmission 
through pipes with bends, branches and discontinuities has been 
studied, which has led to useful design rules concerning the 
position and size of pipe supports for minimum power 
transmission[l]. However, in many practical structures 
transmission paths are composed of more complex curved 
elements. Therefore, there is a need to extend power transmission 
analyses to this class of structure. 


1163 


Wave motion in a curved beam with a constant radius of 
curvature has been considered by Love [2] who assumed that the 
centre-line remains unextended during flexural motion, whilst 
flexural behaviour is ignored when considering extensional 
motion. Using these assumptions the vibrational behaviour of 
complete or incomplete rings has been considered by many 
researchers who are interested in the low frequency behaviour of 
arches and reinforcing rings. In reference [2] Love also presented 
equations for thin shells which include the effects of extension of 
the mid-surface during bending motion. Soedel [3] reduced these 
equations and made them applicable to a curved beam of constant 
radius of curvature. In an alternative approach Graff [4] derived 
these equations from first principles and also constructed frequency 
verses wavenumber and wavespeed versus wavenumber graphs. 
Philipson [5] derived a set of equations of motion which included 
extension of the central line in the flexural wave motion, and also 
rotary inertia effects. In a development analogous to that of 
Timoshenko for straight beams, Morley [6] introduced a correction 
for radial shear when considering the vibration of curved beams. 
Graff later presented frequency versus wave number and wave 
speed versus wave number data for wave motion in a curved beam, 
when higher order effects are included [7]. 

In this paper, expressions for vibrational power transmission 
in a curved beam are derived from first principles. In the next 
section two sets of governing equations for wave motion in a 
curved beam are presented both of which include coupled 
extensional-flexural motion. The first set is based upon a reduction 
of Love's thin shell equations mentioned above. The second set is 
based upon a reduction of Fliigge’s thin shell equations [8]. In 
section three, the expressions for stresses and displacements 
presented in section two are used to derive formulae for vibrational 
power transmission in terms of centre-line displacements. By 
assuming sinusoidal wave motion, expressions are developed 
which relate the time-averaged power transmission to the 
extensional and flexural travelling wave amplitudes. In section 
four, corrections for rotary inertia and shear deformation are 
introduced. The results of numerical studies of these expressions 
are presented which show the effect upon wave motion and power 
transmission of (i) the degree of curvature, and (ii) the various 
simplifying assumptions made about the beam deformation. 


1164 


2. WAVE MOTION IN CURVED BEAMS 


In this section the governing relations between displacements, 
strains, stresses and force resultants in a curved beam are presented. 
The centre-line of the beam lies in a plane and forms a constant 
radius of curvature. The cross-section of the beam is uniform and 
symmetrical about the plane and it is assumed that there is no 
motion perpendicular to the plane. It is also assumed that the beam 
material is linearly elastic, homogeneous, isotropic and continuous. 

Consider a portion of the curved beam, as shown in Figure 1. 
The circumferential coordinate measured around the centre-line is 
s, while the outward pointing normal coordinate from the centre¬ 
line is z, and the general radial coordinate is r. A complete list of 
notation is given in the appendix. For small displacements of thin 
beams the assumptions, known as "Love's first approximation" in 
classical shell theory, can be made [8]. This imposes the following 
linear relationships between the tangential and radial 
displacements of a material point and components of displacement 
at the undeformed centre-line: 


U (r, s, t) = u (R, s, t) + z (|) (s, t) (1) 

W (r, s, t) = w (R, s, t) (2) 

where u and w are the components of displacement at the centre¬ 
line in the tangential and radial directions, respectively, (j) is the 
rotation of the normal to the centre-line during deformation: 


(|> 



3w 


/ angle of'X {rotational displacement^ 

Vcurvature/ \ of straight beam ) 


(3) 


and W is independent of z and is completely defined by the centre¬ 
line component w. 

Circumferential strain consists of both an extensional strain 
and bending strain component. Expressions for these are listed in 
table 1. The strain-displacement expressions of the Love and Flugge 
based equations are identical. However, in the total circumferential 
strain of the Love based theory, the term in the denominator 
has been neglected with respect to unity. Assuming the material to 
be linearly elastic, the circumferential stress-strain relationship 
is given by Hooke's Law, whilst the shear strain, Ysr/ and shear 
stress, Osr/ are assumed to be zero. Assuming the material 


1165 


to be homogeneous and isotropic, the material properties E, G and v 
can be treated as constants. Thus, by integrating the stresses over 
the beam thickness, force and moment resultants can be obtained, 
which are listed in table 2. The adopted sign convention is shown in 
Figure 2. 

Equations of motion for a curved beam are presented in [4]. 
These equations are derived in terms of the radian parameter 0. By 
applying the substitution, s - RQ, the equations of motion can be 
expressed in terms of the circumferential length, s. These equations 
are listed in [9] along with the Fliigge based equations of motion 
which have been obtained by a reduction of the equations of 
motion for a circular cylindrical shell presented in [8]. An harmonic 
solution of the equations of motion can be obtained by assuming 
that extensional and flexural sinusoidal waves propagate in the 
circumferential direction. The harmonic form of the equations of 
motion are also listed in [9]. 


3. VIBRATIONAL POWER TRANSMISSION IN CURVED BEAMS 

In this section the expressions for displacements and stresses 
presented in section two are used to derive the structural intensity 
and power transmission due to flexural and extensional travelling 
waves in a curved beam. The structural intensity expressions are 
formulated in terms of displacements at the centre-line. By 
assuming sinusoidal wave motion, expressions are developed 
which relate the time-averaged power transmission to the flexural 
and extensional travelling wave amplitudes. 

Structural Intensity in the circumferential direction of a curved 
beam is given by [10]: 


Is = 


au 
at 

f intensity due to 
I circumferential stress 


aw 
at 

f intensity due to 
radial shear stress 


( 4 ) 


By integrating across the beam thickness power transmission per 
unit length in the circumferential direction is obtained: 

h/2 

Ps= J Isdz (5) 

-h/2 


1166 


Substituting the Love based circumferential stress-strain relation 
and strain-displacement expression into equation (5) the power 
transmission due to circumferential stress is obtained. (A full 
derivation is given in [9].) By analogy to power transmission in a 
straight beam [1] this can be expressed in terms of an extensional 
component, Pe, and a bending moment component, Pbm* Although 
the transverse shear stress Ogr is negligible under Love's first 
approximation, the power transmission due to transverse shear 
stress can be evaluated from the non-vanishing shear force, Q, 
because the radial displacement W does not vary across the beam 
thickness. Again, by analogy to power transmission in a straight 
beam [1] this is expressed as a shear force component. Thus, the total 
power transmission in the circumferential direction is given by the 
sum of the extensional, bending moment and shear force 
components. These equations are listed in table 3 along with Fliigge 
based power transmission equations which are also listed in table 3. 
Substituting harmonic wave expressions into the Love and Fliigge 
based power transmission equations gives expressions for the power 
transmission in the circumferential direction in terms of travelling 
wave amplitudes A and B. For sinusoidal wave motion it is useful 
to develop time-averaged power transmission defined by [1]: 


T/2 


<Ps> 


-- f 

t“T J 


Ps (s,t) dt 


- T/2 


( 6 ) 


where T is the period of the signal. Time averaged Love and Fliigge 
based power transmission equations are given in table 4. 


4. THE EFFECT OF ROTARY INERTIA AND SHEAR 
DEFORMATION 

It is known that shear deformation and rotary inertia effects 
become significant for straight beams as the wave length approaches 
the same size as the thickness of the beam, and for cylindrical shells 
as the shell radius decreases [8]. Thus, the objective in this section is 
to establish more complete equations for power transmission in a 
curved beam and to show under what conditions these specialise to 
the simple bending equations presented in section three. 

Rotary inertia effects are included by considering each element 
of the beam to have rotary inertia in addition to translational 
inertia. Equations of motion for a curved beam which include the 


1167 


effect of rotary inertia are presented in [7]. These equations are 
listed in [9] in terms of the circumferential distance parameter, s. 
Equations for vibrational power transmission can be derived in the 
same marmer as described in section three. These equations are 
listed in tables 3 and 4 where it can be seen that the extensional and 
bending moment components when including rotary inertia effects 
are identical to the corresponding Fliigge based expressions. 
However, the shear force component now contains an additional 
rotary inertia term. 

If shear deformation is included then Kirchoff s hypothesis is 
no longer valid, and the rotation of the normal to the centre-line 
during bending, (|), is no longer defined by equation (3) but is now 
another independent variable related to the shear angle, y. 
However, unlike simple bending theory, where the transverse 
shear strain, Ysr^ is negligible, the transverse shear strain is now 
related to the shear angle, y which is expressed in terms of 
displacements u, w and (j). The circumferential force, bending 
moment, and shear force obtained from [8] are given in table 2. A set 
of equations of motion for a curved beam which includes the effect 
of shear deformation is presented in [9]. Power transmission 
equations in the circumferential direction can be obtained in a 
manner analogous to that used for Love and Fliigge based theories. 
As before, the power transmission due to circumferential stress 
can be identified as consisting of extensional and bending moment 
components. The contribution to the power transmission from the 
transverse shear stress is obtained from the product of the shear 
force resultant and the radial velocity which gives the shear force 
component of power transmission. 


5. NUMERICAL STUDY 

For a given real wavenumber, k, the harmonic equations of 
motion were solved to find the corresponding circular frequency, co, 
and complex wave amplitude ratio. The simulated beam was 
chosen to have the physical dimensions and material properties of 
typical mild steel beams used for laboratory experiments. Four 
different radii of curvature were investigated, which were 
represented in terms of the non-dimensional thickness to radius of 
curvature ratio, h/R. These ratios were ^/lO/100/1000 and 
^/lOOOO- 

Using the Love equations of motion. Figure 3 shows the 
relationship between wave number and frequency for a beam with a 


1168 


thickness to radius of curvature ratio of ^/lo- The frequency range 
is represented in terms of the non-dimensional frequency 
parameter Q = ®^/co, where Cq is the phase velocity of extensional 
waves in a straight bar and the wave number range is represented 
in terms of the non-dimensional wave number, kR. It can be seen 
that two types of elastic wave exist: one involving predominantly 
flexural motion; the other predominantly extensional motion. 
However, for wave numbers less than kR = 1, the predominantly 
flexural wave exhibits greater extensional than flexural motion. 

Solution of the shear deformation equations of motion for a 
curved beam shows that three types of elastic wave exist. These are 
the predominantly flexural and predominantly extensional waves 
of simple bending theory and additionally a predominantly 
rotational wave related to the shear angle. The relationship 
between wave number and frequency for these three wave types is 
shown in Figure 4. 

A numerical investigation of the power transmission 
equations was undertaken using simulated beams with the same 
dimensions and material properties as those used in the previous 
study of wave motion. Figure 5 shows the relationship between 
transmitted power ratio and frequency. For the predominantly 
flexural wave the time-averaged transmitted power ratio is 
calculated by dividing the time-averaged power transmitted along a 
curved beam by a predominantly flexural wave by the time- 
averaged power transmitted by a pure flexural wave travelling in a 
straight Euler-Bernoulli beam. i.e. the ratio (<Pe>t + <Fbm>t + 
<Psf>t)/EIcokf3Af. For the predominantly extensional wave the 
transmitted power ratio is calculated by dividing the time-averaged 
power transmitted along a curved beam by a predominantly 
extensional wave by the time-averaged power transmitted by a pure 
extensional wave in a straight rod. i.e. the ratio (<Pe>t + <Pbn>t + 
<Psf >t)/EScokex P^ex 

6. SUMMARY AND CONCLUSIONS 

In this paper, starting from first principles, expressions for 
vibrational power transmission in a curved beam have been 
derived using four different theories. Love’s generalised shell 
equations include extension of the centre-line during bending 
motion were the first set of equations considered. Fiiigge's 
equations also include centre-line extensions and were the second 
set of equations used. Corrections for rotary inertia and shear 
deformation produced the third and fourth sets of governing 


1169 


equations, respectively. By letting the radius of curvature, R, tend 
to infinity these equations reduce to the corresponding straight 
beam expressions presented in [1]. 

Using the governing equations for each theory, expressions 
were then developed which related time-averaged power 
transmission to the amplitudes of the extensional, flexural and 
rotational displacements. For each theory the effects of curvature 
upon the resulting wave motion and power transmission were 
then investigated using beams with different degrees of curvature. 
From the results of this study it can be seen that vibrational power 
transmission in curved beams can be classified into three different 
frequency regions: 

(i) below the ring frequency, Q = 1, curvature effects are 
important; 

(ii) above the ring frequency but below the shear wave cut-on 

frequency, = 1 the curved beam behaves essentially as a 

straight beam; 

(iii) above the shear wave cut-on frequency, higher order effects 
are important. 


ACKNOWLEDGEMENT 

The analytical work presented in this paper was carried out while 

both authors were at the Institute of Sound and Vibration Research, 

University of Southampton. The financial support of the Marine 

Technology Directorate Limited and the Science and Engineering 

Research Council is gratefully acknowledged. 

REFERENCES 

1. J.L. HORNER 1990 PhD thesis, University of Southampton 
Vibrational power transmission through beam like structures. 

2. A.E.H. LOVE 1940 Dover, Nezv-York. A treatise on the 
mathematical theory of elasticity. 

3. W. SOEDEL 1985 Dekker, New York. Vibrations of shells and 
plates 

4. K.F. GRAFF 1975 Clarenden Press, Oxford. Wave motion in 
elastic solids 

5. L.L. PHILIPSON 1956 Journal of Applied Mechanics 23, 364- 
366. On the role of extension in the flexural vibrations of rings. 

6. L.S.D. MORLEY 1961 Quarterly Journal of Mechanics and 
Applied Mathematics 14, (2), 155-172. Elastic waves in a 
naturally curved rod. 


1170 


7. K.F. GRAFF 1970 IEEE Transactions on Sonics and Ultrasonics, 
SU-17 (1), 1-6. Elastic wave propagation in a curved sonic 
transmission line. 

8. A.W. LEISSA 1977 NASA SP-288, Washington DC Vibrations 
of shells. 

9. S.J. WALSH 1996 PhD thesis, University of Southampton. 
Vibrational power transmission in curved and stiffened 
structures. 

10. A.J. ROMANO, P.B. ABRAHAM, E.G. WILLIAMS 1990 Journal 
of the Acoustical Society of America 87. A Poynting vector 
formulation for thin shells and plates, its application to 
structural intensity analysis and source localization. Part I: 
Theory. 


APPENDIX: NOTATION 

A flexural wave amplitude 

Af amplitude of a purely flexural wave 

B extensional wave amplitude 

Bex amplitude of a purely extensional wave 

C rotation wave amplitude 

E Young’s modulus 

G shear modulus 

I second moment of area of cross-section of beam 

Ig structural intensity in circumferential direction 

K radius of gyration 

M bending moment on cross-section of beam 

N circumferential force on cross-section of beam 

P transmitted power 

Q shear force on cross-section of beam 

R radius of curvature 

S cross-sectional area of beam 

T period of wave 

LF displacement in circumferential direction 

W displacement in radial direction 

Co wavespeed of extensional waves in a straight bar 

Cs wavespeed of shear waves in a straight bar 

ds length of elemental slice of curved beam 

eg total circumferential strain 

h thickness of beam 

k wavenumber 

kex wavenumber of a purely extensional wave 

kf wavenumber of a purely flexural wave 

r coordinate in radial direction 


1171 


s coordinate in circumferential direction 

u displacement at centre-line in circumferential direction 

w displacement at centre-line in radial direction 

z coordinate of outward pointing normal 

Q. non-dimensional frequency 

Ps bending strain 

Y shear angle 

Ysr transverse shear strain 

Ej. radial strain 

£g circumferential strain 

0A phase angle of flexural wave 

0C phase angle of rotational wave 

K Timoshenko shear coefficient 

?iex wave length of extensional waves in a straight bar 

V Poisson's ratio 

Oj. radial stress 

Gg circumferential stress 

0 SJ transverse shear stress 

d change in slope of normal to centre-line 

CO radian frequency 


W(r,s,t) 



1172 




Figure 2: Sign convention and force resultants on an elemental 
slice of curved beam 



Figure 3: Wave number v. frequency relationship for a curved 
beam predicted using Love theory 





Transmitted power ratio (Curved beam)/(Slraight beam) ^ Non-dimensional wavenumber 








Table 1 : Displacement, strain-displacement and stress-strain equations for a curved beam. 



1175 








































Table 2: Force resultants for a curved beam. 



1176 




















Table 3: Power transmission for a curved beam. 



1177 






















Table 4 : Time-averaged power transmission by a single harmonic wave 



1178 

















A Parameter-based Statistical Energy 
Method for Mid-frequency 
Vibration Transmission Analysis 

Sungbae Choi, Graduate Student Research Assistant 
Matthew P. Castanier, Assistant Research Scientist 
Christophe Pierre, Associate Professor 
Department of Mechanical Engineering and Applied Mechanics 
The University of Michigan 
Ann Arbor, MI 48109-2125, USA 

Abstract 

Vibration transmission between two multi-mode substructures con¬ 
nected by a spring is investigated. A classical Statistical Energy 
Analysis (SEA) approach is reviewed, and it is seen that some typ¬ 
ical assumptions which are valid at high frequencies lose accuracy 
in the mid-frequency range. One assumption considered here is 
that of an identical probability density function (pdf) for each reso¬ 
nant frequency. This study proposes a Parameter-based Statistical 
Energy Method (PSEM) which considers individual modal informa¬ 
tion. The results of PSEM have good agreement with those of a 
Monte Carlo technique for an example system. 

Nomenclature 


E[ ] 

expected value 


power transmitted between substructure i and k 

IIfc(u;) . 

total power transmitted to substructure k 

Pi 

power input to substructure i 

Vi, 

coupling loss factor (CLF) 

fh 

modal driving force for mode j of substructure 1 

UJ 

frequency [rad/sec] 

l.l.-.lj.Ol 

subscripts for decoupled Bar 1 

2,2r,2„02 

subscripts for decoupled Bar 2 

El, ^2 

blocked energy 

Eoi, Eo2 

Young’s modulus 

Pl^P2 

density 

mi,m2 

mass per unit length 

Mu M 2 

total mass 

Cl,C2 

viscous damping ratio 

Ai, A 2 

cross-sectional area 

0 

rH 

0 

nominal length 


1179 


Ti,T2 

disordered length 

£1,^2 

ratio of disorder to nominal length 

k 

coupling stiffness 


coupling ratio 

Xi,X2 

position coordinate 

ai,a2 

point-coupling connection position 

Wi, W2 

deflection 


modal amplitude 


mode shape function 

NuN2 

number of resonant frequencies 


resonant frequencies 


lower limit of resonant frequencies 


upper limit of resonant frequencies 

(Tl, (72 

standard deviation of disorder 


1 Introduction 

Vibration transmission analysis between connected substructures in the 
mid-frequency range is often a daunting prospect. Since the analysis at 
high frequencies requires greater model discretization, the size and com¬ 
putational cost of a full structure model (e.g., a Finite Element model) 
can become prohibitive. Also, as the wavelengths approach the scale of 
the structural variations, uncertainties (tolerances, defects, etc.) can sig¬ 
nificantly affect the dynamics of the structure. Starting at what may be 
called the mid-frequency range, deterministic models fail to predict the 
response of a representative structure with uncertainties. 

Therefore, in the mid-frequency range, a statistical analysis of vibra¬ 
tion transmission may be more appropriate. This approach is taken in 
the procedure known as Statistical Energy Analysis (SEA) [1]. In SEA, 
a structure is divided into coupled substructures. It is assumed that each 
substructure exhibits strong modal overlap which makes it difficult to dis¬ 
tinguish individual resonances. Therefore, the resonant frequencies are 
treated as random variables, each with an identical, uniform probability 
density function (pdf) in the frequency range of interest. This assumption 
greatly simplifies the evaluation of the expected value of transmitted vi¬ 
bration energy. A simple linear relation of vibration transmission between 
each pair of substructures is retrieved. The power transmitted is propor¬ 
tional to the difference in the average modal energies of the substructures. 
This relation is analogous to Fourier’s law of heat transfer [1-4]. 

In the low- to mid-frequency region, the modal responses are not strongly 
overlapped. In this case, two typical SEA assumptions are less accurate: an 
identical pdf for all resonant frequencies, and identical (ensemble-averaged) 


1180 


values of the associated mode shape functions at connection positions. In 
this paper, these two assumptions are relaxed. A distinct uniform pdf 
is applied for each resonant frequency, and a piecewise evaluation of the 
transmitted power is performed. This is called a Parameter-based Sta¬ 
tistical Energy Method (PSEM) because it considers the statistical char¬ 
acteristics of individual system parameters. This solution can accurately 
capture peaks of transmitted power while maintaining the SEA advantage 
of efficiency. 

This paper is organized as follows. In section 2, we briefly review SEA 
along with the associated assumptions and limitations. In section 3, the 
power transmitted between two spring-coupled multi-mode substructures 
is investigated by applying several SEA assumptions. A Monte Carlo solu¬ 
tion is used for comparison. In section 4, the PSEM approach is presented 
and the results are shown. Finally, section 5 draws conclusions from this 
study. 

2 Overview of SEA 

In Statistical Energy Analysis, the primary variable is the time-averaged 
total energy of each substructure. This is called the blocked energy, where 
blocked means an assumed coupling condition. The assumed coupling con¬ 
dition may be the actual coupling, a clamped condition at the substructure 
junctions, or a decoupled condition [2, 5]. 

In order to predict the average power transmitted between two directly- 
coupled substructures, a few simplifying assumptions are applied. Some of 
the essential SEA assumptions are summarized by Hodges and Woodhouse 
in Ref. [3]; 

• Modal incoherence: the responses of two different modal coordinates 
are uncorrelated over a long time interval 

• Equipartition of modal energy: all modes within the system have the 
same kinetic energy 

The above conditions make it possible to treat all modal responses as sta¬ 
tistically identical. The first assumption implies a broad band, distributed 
driving force (often called ”rain on the roof’) which leads to uncorrelated 
modal driving forces. The second assumption implies that the substruc¬ 
tures have strong modal overlap, or that the parameter uncertainties are 
sufficiently large that the modes are equally excited in an ensemble average 
sense. Thus, the resonant frequencies are treated as random variables with 
identical, uniform probability density functions (pdfs) for the frequency 
range of interest. 

The SEA relation for the expected value of power transmitted from 


1181 


substructure i to substructure k, may be expressed as 

E[nijfc(cj)] = 0)7},^Ni{^ (1) 

where uj is the frequency, 77 ^^ is the coupling loss factor, Ei is the blocked 
energy of substructure z, and Ni is the number of participating modes of 
substructure i for the frequency range of interest. The power dissipated by 
substructure i is expressed as 

= u:riiEi (2) 

where r]i is the damping loss factor. Using Eqs.(l) and (2), the equation 
of power balance for substructure i at steady state [1, 2, 5, 6] is 

N N 

Pi = E[Ilij] + UJTJiEi = UJ ( 3 ) 

i=i 

j¥i 

where Pi is the power input to substructure i from external sources. Note 
that the first term on the right-hand side is the power transmitted through 
direct coupling between substructures. 


3 Vibration Transmission in a Two-Bar 

System 

The longitudinal vibration of the structure shown in Fig.l is considered 
in this study. The structure consists of two uniform bars with viscous 
damping which are coupled by a linear spring of stiffness k. The spring is 
connected at intermediate points on the bars, Xi = ai and X 2 = 0 , 2 . Bar 
i has nominal length Loi. A parameter uncertainty may be introduced by 
allowing the length to vary by a small random factor e^, which is referred 
to as disorder. The length of a disordered bar is Li = Zrox(l + £:)• The ratio 
of the connection position to the length, aifLi^ is held constant. Bar 1 is 
excited by a distributed force Fi{xi,t). 

3.1 Nominal transmitted power 

The power transmitted from Bar 1 to Bar 2 for the nominal system (no 
disorder) is briefly presented here. A more detailed derivation is shown in 
the Appendix (see also Refs. [7] and [8]). The equations of motion are 

Wi(a:i,t) -f ^Ci(wi(a;i,f)) = Fi(a:i,t)-|- 

^[W2(a2, t)-wi(ai, - ci) 

(jBo2A2-§^ + m2|^^W2(x2,t) + ^C'2(w2(a;2,t)) = 

k[wi{ai,t)-W2{a2,t)]6{x2 - ^ 2 ) 


1182 


a) 

Bar 2 


Fig. 1: Two-bar system 

where ^ is a Dirac delta function, and (for Bar i) Eoi is Young’s modulus, 
Ai is the cross-sectional area, rm is the mass per unit length, ’Ci is the 
viscous damping operator, and Wt(a:i,t) is the deflection. The deflections 
of the two bars can be expressed by a summation of modes: 

Wi{xi,t) = W2(x2,t) = E 1^2r(i)^2r(^2) (5) 

t=0 r=0 

where and W 2 ^{t) are modal amplitudes, and and ^ 2 r(^ 2 ) 

are mode shape functions of the decoupled bars. These mode shape func¬ 
tions are normalized so that each modal mass is equal to the total mass of 
the bar, Mi. Applying modal analysis and taking a Fourier transform, the 
following equations are obtained: 

Mi(f)ijWij = 4- fc^ij(ai)[E kF2r^2r(<^2) - E 

_ (6) 

M2<f>2,W2, = ^^^2,(<22)[E VFi,^i,(ai) - E W2.^2M2)] 

i=0 r=0 

[ul. - -f • 2Cia;i .a;)(2 - sgn(j)) 
y/^ • 2 C 2 <^ 2.^)(2 - sgn(s)) 

1 for ^ > 0 
0 for 2 = 0 

where an over-bar (") denotes a Fourier transform, and are resonant 
frequencies, (i and (2 are damping ratios, and is a modal driving force. 
Mode 0 is a rigid body mode, which is why the sgn(i) term is present. 
Note that the damping ratio of each bar is assumed to be the same for all 
modes. 

Next, the modal driving forces are assumed to be incoherent, and each 
spectral density function is assumed to be constant (white noise) over a 
finite range of frequency [2]. After some algebra, the power transmitted 


<^2^ = 

sgn(0 = 



1183 



(7) 


from Bar 1 to Bar 2, 1112 ( 0 ;), is found as: 

2o;^PC2«^PlPl ^ ^2r(^2)0^2r 

M1W2IAI2 ^ |(^ 2.|2 

-^1 -^2 <?^ 2 r 

where 5pjpi is the same uniform spectral density function for each modal 
driving force on Bar 1. 


1112(0;) = 
A = 


3.2 Monte Carlo Energy Method (MCEM) 

The disordered case is now considered, where each bar has a random length. 
The ensemble-averaged transmitted power for a population of disordered 
two-bar systems is found by tahing the expected value of Eq. (7): 


^[ni2] 




( 8 ) 


where Ni is the number of modes taken for Bar i (this is an arbitrary set of 
modes that have been aliased to the numbers 1,2,- • ’,Ni). Since a truncated 
set of modes is used, Eq. (8) is an approximation. The random variables 
in Eq. (8) are the resonant frequencies of the bars (which are present in 
the terms ^i;, ^ 2 r? s,nd A). 

Equation (8) may be solved numerically using a Monte Carlo method: 
the random variables are assigned with a pseudo-random number generator 
for each realization of a disordered system, and the transmitted power 
is averaged for many realizations. This is called a Monte Carlo Energy 
Method (MCEM) here. It may be used as a benchmark for comparing the 
accuracy of other approximate methods. 

Note that the resonant frequencies of a bar may be found directly from 
the disordered length. Therefore, for the MCEM results in this study, the 
actual number of random variables in Eq. (8) is taken to be one for each 
decoupled bar. That is, the two random lengths are assigned for each real¬ 
ization, and then the natural frequencies are found for each bar in order to 
calculate the transmitted power. If such a relation were not known, each 
resonant frequency could be treated as an independent random variable. 


3.3 SEA-equivalent Transmitted Power 

An SEA approximation of the transmitted power may be obtained by ap¬ 
plying several typical SEA assumptions to Eq. (8). (Since Eq. (1) is not 
used directly, this might be called an SEA-equivalent transmitted power.) 
These assumptions were summarized in Ref. [8]: the coupling between 


1184 


substructures is weak, the modal responses are uncorrelated, the expected 
value of the square of mode shape functions at connection positions is unity, 
and the pdfs of the resonant frequencies are uniform and identical. 

The assumption of weak coupling means that the value of jAj in Eq. 
( 8 ) is approximately one. Applying the second and third assumptions then 
yields 


E[ai2] 


M?M2 









(9) 


Since the pdf of each resonant frequency is assumed to be uniform, the 
expected values in Eq. (9) are 


E[ 



E[ 



l/4u)" 

1 


. 2 cos| 


-f 2 a;a;i. • cosf + 
^ ul. — 2u}ujii • cos| • 

• sin I 


sin% 


ian~ 


U) 


2 _ 






( 10 ) 


l/4a;"C2Vr^, - 1) 

- 1 - tan ——--— 


W 2 r =‘*' 2 ru 


— ^2r 


4w^C2\/i - Cl 


(11) 


W2r=‘*'2rj 


where a = cos~^(l - 2 Cj), subject to the restrictions (1 - 2(1)^ < 1 and 
(1 _ 2 ( 2)2 ^ I Finally, since the pdfs of the resonant frequencies are taken 
to be identical, the frequency limits do not depend on the individual modes 
(a;i. = uj 2 r, - a^nd = u; 2 ,„ = oju)- Therefore, each sum in Eq. (9) 
simplifies to the product of the expected value and the number of modes 
in the frequency range of interest: 


E[n^2] 


2C2A:ViViiV25,^p^ r.r 1 1 Pr ^2. ^ 

WM2 Vi.r 


( 12 ) 


Equation (12) is the SEA approximation used in this study. 


3.4 Example 

The three formulations of the transmitted power presented thus far the 
nominal transmitted power in Eq. (7), the MCEM transmitted power in 
Eq. ( 8 ), and the SEA-equivalent transmitted power in Eq. (12) — are 
now compared for a two-bar system with the parameters shown in Table 1. 
For the MCEM results, the disorder (ei and 62 ) was taken to be uniformly 
distributed with mean zero and standard deviation ai = 0-2 = 10%. 

As a measure of the coupling strength, the coupling ratio, Ri, is defined 
cLS the ratio of coupling stiffness to the equivalent stiffness of a bar at the 


1185 


Table 1: Material properties and dimensions of two bars 


MIIM 2 

21.53/21,53 

[Kg] 

R 01 IL 02 

10.58/8.817 

[m] 

E 01 IR 02 

200 XIOV2OO xlO^ 

[N/m'1 

ai/a2 

2.116/7.053 

N 

Pllp2 

7,800/7,800 

[Kg/m'*l 

^pipi 

1 

[N^l 

C1/C2 

0.005/0.005 


k 

4.868 xlO^ 

[N/m; 



Fig. 2: Comparison of the nominal transmitted power, the MCEM results 
(20,000 realizations with (j^ = <72- 10%), and the SEA approximation. 


fundamental resonant frequency, Ri = weak cou¬ 

pling is considered here such that Ri = 0.01. 

The nominal transmitted power, the MCEM results, and the SEA ap¬ 
proximation are shown in Fig.2. The transmitted power calculated for 
the nominal system exhibits distinct resonances. This is due to the low 
modal overlap of the bars in this frequency range. The MCEM results 
show distinct peaks for uj < 15,000 rad/s, but they become smooth as 
the frequency increases. The SEA approximation does not capture indi¬ 
vidual resonances. However, at the higher frequencies where the disorder 
effects are stronger, the SEA approximation agrees well with the MCEM 
results. The frequency range between where the MCEM results are close 
to the nominal results and where they are close to the SEA results (ap¬ 
proximately 2,500 ~ 15,000 rad/s for this case) is considered to be the 
mid-frequency range here. This range will vary depending on the system 


1186 











parameters and the disorder strength. In the next section, an efficient ap¬ 
proximation of the transmitted power is presented which compares well 
with MCEM in the mid-frequency range. 


4 Parameter-based Statistical Energy 
Method (PSEM) 

The SEA approximation presented in the previous section does not capture 
the resonances in the transmitted power because of two assumptions: the 
resonant frequencies all have the same uniform pdf, and the values of the 
square of mode shape functions at the connection positions are taken to be 
the ensemble-averaged value. Keane proposed an alternate pdf of resonant 
frequencies in order to apply SEA to the case of two coupled nearly periodic 
structures [9]. This pdf is shown in Fig.3(a). It accounts for the fact that 




Fig. 3: (a) The pdf of the natural frequencies and the resultant transmitted 
power from Ref. [9]. (b) The pdfs of three natural frequencies, and a 
schematic representation of the piecewise evaluation of transmitted power 

for PSEM. The individual modal contributions are extrapolated (-) and 

summed to calculate the total transmitted power (—). 

the natural frequencies of a nearly periodic structure tend to be grouped in 
several distinct frequency bands. Thus the pdf has a large constant value 
for those frequency bands, and a small constant value elsewhere. The SEA 
approximation of transmitted power is then modified by simply adding a 
positive value or negative value on a logarithmic scale, as demonstrated in 
Fig, 3(a). This solution thus captures some of the resonant behavior of the 
transmitted power. 


1187 












Here, a more general approach is taken for approximating the power 
transmitted between two substructures in a frequency range in which they 
have low or intermediate modal overlap. Each resonant frequency is as¬ 
signed a uniform pdf. However, the frequency range of each pdf is diiferent; 
it corresponds to the range in which that resonant frequency is most likely 
to be found. (The concept of using “confidence bands” as one-dimensional 
pdfs was suggested but not pursued in Ref. [8].) An example is shown in 
Fig. 3(b) for three resonant frequencies. Furthermore, it is assumed that 
the values of the square of the mode shape functions at the connection 
positions are known. Thus, applying only the first two SEA assumptions 
along with those noted above, Eq. (8) becomes: 




N2 

i:kme 


This is called a Parameter-ba.sed Statistical Energy Method (PSEM) be¬ 
cause it employs information for individual modal parameters. 

Since each modal pdf is uniform, Eqs. (10) and (11) still hold for the 


expected values in Eq. (13). However, unlike the SEA approximation, each 
expected value is different, because the corresponding frequency bounds are 
unique. Furthermore, note that the pdfs do not cover the entire frequency 
range of interest. The results for each mode are therefore extrapolated 
outside the frequency range of that modal pdf before the individual modal 
contributions are summed. This is shown schematically in Fig. 3(b). PSEM 


is therefore a piecewise evaluation of the expected value of transmitted 


power. 

The PSEM approximation is now applied to the two-bar system of 
Table 1, with the standard deviation of disorder cti = 0-2 = 10%. The pdfs 
of the resonant frequencies of Bar 1 and Bar 2 aze shown in Fig. 4(a) and 
(b), respectively. For this system, the bounds for each resonant frequency 
may be found directly from the variation of the uncertain parameter. It 
can be seen that the spread of each natural frequency pdf due to disorder 
increases with increasing frequency. 

The MCEM, PSEM, and SEA approximations for the transmitted power 
are shown in Fig. 4(c). There were 20,000 realizations taken for the MCEM 
results at each sampled frequency. This took about 10 hours of computa¬ 
tion time. In contrast, the PSEM results only required 3 seconds of compu¬ 
tation time, and the SEA results only required about 1 second. Note that 
the PSEM results show excellent agreement with the much more expensive 
MCEM results. The difference at very low frequencies comes from the fact 
that for the PSEM approximation, the value of the term |A| was assumed 
to be one due to weak coupling. This assumption breaks down as the 
frequency approaches zero. However, the match between the MCEM and 


1188 


Fig. 4: (a) Natural frequency pdfs for Bar 1. (b) Natural frequency pdfs 
for Bar 2. (c) Transmitted powers obtained by MCEM, PSEM, and SEA 
for c7i = <J2 = 10%. 

PSEM results in the mid-frequency range is excellent. Again, it is noted 
that the SEA results converge to those of MCEM (and PSEM) as the fre¬ 
quency increases. Now it can be seen that the assumption of identical pdfs 
for all modes becomes better with increasing frequency. 

Next, the example system is considered with smaller disorder, Ci = 
(72 = 1%. Fig. 5 shows the results for this case for what might be called 
the mid-frequency range. Note that even though this is a higher frequency 
range than that considered for the previous ca.se, the pdfs of the resonant 
frequencies shown in Fig. 5(a) and (b) are not as strongly overlapped. 
Thus, several peaks are seen in the transmitted power in Fig. 5(c). Again, 
the PSEM approximation agrees well with the MCEM results, although 
there is more discrepancy for this case. The SEA approximation follows 
the global trend, but does not capture the resonances or anti-resonances. 
The SEA results drop off at the edges because only modes within this 
frequency range are considered to contribute to the transmitted power. 

In addition to PSEM, another piecewise evaluation of the transmitted 
power is considered here. For this approximation, wherever the individual 
mode pdfs overlap, they are superposed to form a pdf for all the modes in 
that “section” of the frequency range. This superposition is demonstrated 
in Fig. 6. Also, if the number of modes in a section is above a certain cutoff 
number, Nc, then it is assumed that their mode shape function values at 
the connection positions are unknown, so that the ensemble-averaged value 
must be used. This is called a multiple mode approximation. The purpose 


1189 









Fig. 5: (a) Natural frequency pdfs for Bar 1. (b) Natural frequency pdfs 
for Bar 2. (c) Transmitted power obtained by MCEM, PSEM,-and SEA 
for cTi = 0-2 = 1 %. 



Fig. 6: Resonant frequency pdfs for PSEM and for the multiple mode 
approximation. 

of formulating this approach is to investigate what happens as information 
about the individual modes is lost. 

The multiple mode approximation is applied to the example system 
with cTi = <72 = 10% in Fig.7. For A/c = 2, this approximation has good 
agreement with MCEM. The match is especially good for u) < 10,000. 
Above this frequency, the number of overlapped resonant frequencies in 
each pdf section is greater than Nc, and the loss of mode shape information 
affects the results slightly. For Nc = 0, the values of the mode shape 
functions are taken to be one for the entire frequency range, just as in the 
SEA approximation. As can be seen in Fig. 7, the piecewise construction 
of the pdf roughly captures the frequency ranges of the resonances and 
anti-resonances. However, the mode shape effect is more pronounced in 


1190 




















-B.S 


- MCEM 

-e.s — Multiple mode approximation rvt= 2 

. Multiple mode approximation Nfc= O 



Fig. 7: Transmitted power obtained from MCEM and the multiple mode 
approximation for Ui = <72 = 10% 

the mid-frequency range. The peak values are now similar to the SEA 
approximation. 


5 Conclusions 

In this study, the power transmitted between two multi-mode substruc¬ 
tures coupled by a spring was considered. A Monte Carlo Energy Method 
(MCEM) was used to calculate the ensemble average of the transmitted 
power for the system with parameter uncertainties. A classical Statistical 
Energy Analysis (SEA) approximation matched the Monte Carlo results 
in the high-frequency range, but did not capture the resonant behavior of 
the transmitted power in the mid-frequency range where the substructures 
have weak modal overlap. 

A Parameter-based Statistical Energy Method (PSEM) was presented 
which uses a distinct pdf for each natural frequency as well as some indi¬ 
vidual mode shape information. A piecewise evaluation of the transmitted 
power was performed, and then the modal contributions were extrapolated 
and superposed. The PSEM approximation compared very well with the 
much more expensive Monte Carlo results, including in the mid-frequency 


1191 








range. 


References 


1 R. H. Lyon. Statistical Energy Analysis of Dynamical Systems: Theory 
and Applications. M.I.T. Press, 1st edition, 1975. 

2 R. H. Lyon. Theory and Application of Statistical Energy Analysis. 
Butterworth-Heinemann, 2nd edition, 1995. 

3 C. H. Hodges and J. Woodhouse. Theories of noise and vibration trans¬ 
mission in complex structures. Rep.Prog.Physics, 49:107-170, 1986. 

4 J. Woodhouse. An approach to the theoretical background-of statis¬ 
tical energy analysis applied to structural vibration. Journal of the 
Acoustical Society of America, 69(6):1695-1709, 1981. 

5 M. P. Norton. Fundamentals of Noise and Vibration Analysis for En¬ 
gineers. Cambridge University Press, 1st edition, 1989. 

6 N. Lalor. Statistical energy analysis and its use as an nvh analysis tool. 
Sound and Vibration, 30(l):16-20, 1996. 

7 Huw G. Davies. Power flow between two coupled beams. Journal of 
the Acoustical Society of America, 51(1):393-401, 1972. 

8 A. J. Keane and W. G. Price. Statistical energy analysis of strongly 
coupled systems. Journal of Sound and Vibration, 117(2):363-386, 
1987. 

9 A. J. Keane. Statistical Energy Analysis of Engineering Structures 
(Ph.D Dissertation). Brunei University (England), 1988. 


Appendix 


In this appendix, the nominal transmitted power in Eq. (7) for the mono- 
coupled two-bar system is derived. The procedure follows that of Refs [7, 
8 ]. 

Plugging Eq. (5) into Eq. (4), 


F^ixut) + fcf E - Oi) (A.1) 


t=0 


1192 




k £ Wi,(<)$i,(a,) - £ lV2,(«)«2,(a2) 5 (x 2 - aj) 


Multiplying Eq. (A.l) by and integrating with respect to xi for 

[0, Li] yields 


MiK + = 

A, + ff; W2,(4)W2 Xo2) - E 


Li 

2CijUijMi = y C'i(«'i,.(a:i))Wij(xi)cia:i , fi, J Fi{xi,t)^i.{xi)dx], 

0 0 

and wij is the ;th resonant frequency of decoupled Bar 1. The damping 
ratio in Eq. (A.3) is now assumed to be the same (Ci) for all modes, since 
the differences in the ratio are usually small and this simplifies the equation. 
Taking the Fourier transform of Eq. (A.3) with zero initial conditions leads 
to the following 

= 7i,+ S:$i,(ai)fETr2,$2,(02) 

t r=0 i=0 -* 

</>!;• = K, - • 2(iu;i^iv){2 - sgn(i)) 

J fori = 0 

where (~) denotes a Fourier transform. Similarly, applying the previous 
procedure to Eq. (A.2), 

r OO CO -1 

M2(^2.1V2. = ^^$2.(02) EW"li®l.(<»0-EW"2,$2,(o2) (A.5) 

t i=0 r=0 

^ 2 s = {^ 2 , - • 2 C 2 ^^ 2 ,w )(2 - sgn(s)). 

Solving for VF 2 , from Eq. (A.4) and (A.5), 

= Trr-Wx-l^^- - (a.6) 


where 


_ W2.(a2).^/ii$i,(ai) 




1193 


Calculating the second term in brackets in Eq. (A.6), 
fc4'2.(<22)EW^2,«'2.(a2) E 


where 


Plugging Eq. (A.7) into Eq. (A.6), 

M2<j>2s 1 + CKl + Q;2 1=0 

\ I' I. -I — ✓ 

Coupling force Ps 

Using the definition of transmitted power in Ref. [7], 1112 ( 0 )) is 


ni2 = Re[-v^o)f;E[P.F;,(o))] 

L „_ri 


"I" 0^1 + P i~0 j=:0 J 

A 

i<?^2.p hh Ml, 

where Re[] denotes the real part of the argument, * is a complex conjugate, 
and 

Finally, it is assumed that the modal driving forces, f-^., are uncorrelated. 
Also, the spectral density function of each modal driving force is assumed 
to be constant for the finite frequency range of interest: 




C ( \ f ^PlVl ^ 

V for i^j. 


Therefore, Eq. (A.9) becomes 


2^2fc^O)^.5^pipi ^ ^ ^2r(^2)o)2r 

to to 


(A.IO) 


(A.ll) 


1194 


PASSIVE AND ACTIVE CONTROL III 



Research on Control Law of Active Siispension of a 
Seven Degree of Freedom Vehicle Model 

Dr&Prof. YuchengLei Lifen Chen 
Automobile Engineering Dept, Tong Ji University ,Shang 
Hai,P.R.of China 

Abstract 

In the paper , control law of active suspension is presented , 
which involves 7-DOF vehicle model for improving control 
accuracy .The control law involve vehicle running velocity , 
road power spectrum , suspension stiffness and damping .The 
control law can be applied to multi-DOF control of active 
suspension of vehicle . 

Keywords: Active suspension , control law , Game theory , 
Modeling , 7 — DOF Vehicle Model. 

1. Introduction 

An individual control system for each wheel by applying the 
optimum regulation method for the two degrees of freedom is 
showed in [3] . [4] and [5] also introduce two-DOF feedback 
control method of active suspension .It is difficult for two- 
DOF control method to coordinate multi-DOF kinematic 
distances of entire car . Muti-DOF active control can improve 
coordination control accuracy of entire car, but high speed of 
CPU is asked for control and calculation while control law of 
multi-DOF is got by real-time calculation . And, ride 
performance and handling performance is inconsistent . For 
resolving the problem ,the paper holds a new calculation 
method for optimizing the law that can be programmed for 
real-time control by table-lookup and not by real-time 
calculation .So the method and law can not only improve 
coordination control accuracy ,but also develop control 
speed. 

2. Mathematical Model 

Vehicle is simplified to turn into 7 DOF model . 7-DOF 
vibration motion equation can be written as follow 

[Af ]z+ [c]z+ [kY = \C, ](2+ ]e (1) 


1195 


Where [M] is mass matrix , [C] is suspension damping 

matrix , [A’] is suspension stiffness matrix , [C,] is tyre 

• • 

damping matrix , \K, ] is tyre stiffness matrix , Z is 
acceleration matrix ,Z is velocity matrix , Z is 7-DOF 


displacement matrix, Q is road surface input velocity matrix, 
Q is road surface input displacement matrix. 

z=p„z....,zy (2) 


•• . • -j-J 

Where Zi is vehicle vertical acceleration, Zi is roll 

•• •• 

acceleration , Z 3 is pitch acceleration , Z,- (/ = 4,*«*,7) is 

four tyres vertical acceleration . 

3 General Optimization Method of Control Law 
Objective function of optimization of control law can 
generally got by calculating weighted sum of 7-DOF mean 
square root of acceleration , dynamic deflection and dynamic 
load .it can be written as follows 

/=! >1 A =1 

Where a-^ (z = !,• • (7 = !,• • •A), r * = h* • •A) is 

weighted ratio . Where a.. (i=l, * * ' ,7) is 7-DOF mean 

Zi 

square root of acceleration , o-jy_ 0=1? ‘ ' is 7-DOF mean 
square root of dynamic deflection , (k=l, * * ^ A) is 

7-DOF mean square root of relative dynamic load .cr.., 

Zi 

can be calculated by resolving (1) using numerical 

method. 

4 Result of General Optimization 
Method of Control Law 

Optimization result of control law of a truck is got using 
above method as figure 1 and 2 , its main parameters as 
follows. 

Wheel distance is 1.4 meter , axle distance is 2.297 meter , 
mass is 1121.3 kg , front tyre and axle’s mass is 22.8 kg , 
rear tyre and axle’s mass is 35.0 kg , X axis’ rotational 


1196 


inertia is 307.4 kg-ni^ , Y axis’ rotational inertia is 1276.5 
kg-rri^ . 

In fig. 1 and 2 , Cl of RMSMIN and C3 of RMSMIN are 
respectively front and rear suspension damping of getting 
minimization of above objective function , it is changing 
while road surface rough coefficient and automobile 

velocity V is changing . Cl and C3 also rise when velocity V 
rises . This is called control law of general optimization 
method of active suspension in the paper . The result in fig. 1 
and fig. 2 has been verified by road test. 



Fig. 1 front suspension Fig. 2 rear suspension 

optimization damping optimization damping 

Simulation result can also verify that ride performance’s 
increasing (suspension stiffness reducing) will make handling 
performace reduce . So selecting perfect 

= is very difficult 

and inconsistent .The paper advances next game method to 
try to resolve the inconsistent problem . 

5 Game Optimization Method of Control Law 
Because to select weighted ratio of general optimization 
method is difficult , the paper advances a new method of 
optimization of control law — Game Balance Optimization 
Method 4t is discussed as follows . 

Game theory method of two countermeasure aspect can be 
expressed as follows: 


1197 




— 1 j ^ 0 ,z •— 1, 2, 

tr (4) 

^ hj — I , hj ^0 , y = 1, 2,- • *, WZ 2 

where .r^, is probability of selecting R^,R 2 > .of 

countermeasure R ( where R is acceleration mean square 

root ) , and h,,h^, . X, is probability of selecting 

., of countermeasure H ( where H is 

mean square root of deflection or handling and 

satiability ) . 

It is called hybrid game method while these probability is 
leaded into the method. Countermeasure R selects in order 
to get maximization of minimization paying expected value 
of column vector of paying matrix , and countermeasure H 
selects hj in order to get minimization of maximization 
paying expected value of row vector of paying matrix . 


If rank of paying matrix is x ,R should select r., as 
follows. 


max\min\ 






(5) 


relative 


And H should select hj as follows : 


MAXl MINI 






j=i 



( 6 ) 


relative hj 

a.j (z = l,.,y = l,.,^ 2 ) in (5) and (6) is element 

value of paying matrix , basing vehicle theory it can be got 
as follows : 


a.j = C, ! + (7) 

Where C^,C^ in (7) is coefficient of paying matrix ( The 
paper orders they is 1 as an example , as Q ,C 2 ’ s real value 
about very much condition is related to some privacy 


1198 









problem it can’t be introduced.) . (5),(6) called respectively 
minimization maximization expected value and maximization 
minimization expected value can be abbreviated as 
MAxi^IN^^ )} and MIN ^AX(^ )} . if r.,andhj is 

got as optimization of countermeasure , it can be wntten as 
follows : 

MAX^IN^ ^^Optimization Countermeasure 

Expected Value ^ MIN 

A probability association {r^Xj) can be content with 

optimization expected value as follows : 

Optimization Countermeasure Expected 

m^ m-i 

Value=|] J^aij»r;»hj (9) 

/=! 

Writer advances reformation simplex algorithm for resolving 
the game problem as reference [1] . In the paper the water 
selects only an example to introduce calculation results as 
follows because the paper has limited space . 

6 Results of Game Optimization Method of Control Law 
Paying matrix as fig. 3 and fig. 4 , optimization result of 
control law of a truck is got using above game method as 
figure 5 and 6 , calculated truck’s main parameters as 
follows. 

Wheel distance is 1.23 meter , axle distance is 3.6 meter , 
mass is 13880.0 kg , front tyre and axle’s mass is 280.0 kg , 
rear tyre and axle’s mass is 280.0 kg , X axis’ rotational 
inertia is 1935 kg^m^ , Y axis’ rotational inertia is 710 
kg-m- . 

In fig. 3 and 4 3(IJ) is paying matrix value 

{i = 1,. ,m, J = 1,.,m,). In fig. 5 and fig. 6 , K1 of 

RMSMIN and K3 of RMSMIN are respectively front and 
rear suspension stiffness of getting optimization 
countermeasure expected value of above game method , it is 
changing while road surface rough coefficient Q and 

automobile velocity V is changing . K1 and K3 also rise 


1199 




when velocity V rises . This is called control law of active 
suspension . The result in fig. 5 and fig. 6 has been verified 
by road test. 




Fig. 3 paying matrix Fig. 4 paying matrix 

7 Conclusion 

The paper introduces two method to get optimization control 



Fig. 5 front suspension Fig. 6 rear suspension 

optimization stiffness optimization stiffness 

law of active suspension , and the control law is verified to 
ability to be applied to real control of active suspension . This 
will develop control accuracy and speed of active 
suspension .Off course , it need being researched further . 

8 reference 

1 Lei Yucheng , Theory and Engineering Realization of 
Semi — Active Control of Vehicle Vibration ,Dr. paper , 
Harbin Institute of Technology,China ,1995 6 . 

2 Thompson A.G. , A Suspension Proc. Int of Mechanical 


1200 



Engr. Vol 185 No.36,970 — 990,553 563 . 

3 Lei, S., Fasuda , E. and Hayashi, Y. : “An Experimental 
Study of Optimal Vibration Adjustment Using Adaptive 
Control Methods ”, Proc . IMechE Int . Conf . Advanced 
Suspensions , London , England , (1988) , C433/88 , 119- 
124. 

4 Kamopp D. ,Active Damping in Road Vehicle Systems , 
VSD, 12(1983), 291-316. 

5 Kamopp D . C ., Grosby M. J. & Harword R. ,Vibration 
Control Using Semi-Active Force , Generator , Trans . 
ASME, J . Eng . for Ind . Vol. 96 Ser . B , No .2 , (1974), 
619-626. 


1201 


1202 


Designing Heavy Truck Suspensions for Reduced Road Damage 

Mehdi Ahmadian 
Edward C. Mosch Jr. 


Department of Mechanical Engineering 
Virginia Polytechnic Institute and State University 
Blacksburg, Virginia 24061-0238; USA 
(540) 231-4920/-9100(fax) 
ahmadian@vt.edu 


ABSTRACT 

The role of semiactive dampers in reducing tire dynamic loading is examined. 
An alternative to the well-known skyhook control policy, called 
“groundhook,” is introduced. Using the dynamic model of a single suspension, 
it is shown that groundhook semiactive dampers can reduce tire dynamic 
loading, and potentially lessen road damage, for heavy trucks. 


INTRODUCTION 

The main intent of this work is to determine, analytically, the role of 
semiactive suspension systems in reducing tire dynamic loading, and road and 
bridge damage. Although primary suspension systems with semiactive 
dampers have been implemented in some vehicles for improving ride and 
handling, their impact on other aspects of the vehicle remain relatively 
unknown. Specifically, it is not yet known if implementing semiactive 
dampers in heavy truck suspension systems can reduce the tire dynamic 
forces that are transferred by the vehicle to the road. Reducing dynamic forces 
will result in reducing pavement loading, and possibly road and bridge damage. 

The idea of semiactive dampers has been in existence for more than 
two decades. Introduced by Karnopp and Crosby in the early 70’s [1-2], 
semiactive dampers have most often been studied and used for vehicle primary 
suspension systems. A semiactive damper draws small amounts of energy to 
operate a valve to adjust the damping level and reduce the amount of energy 
that is transmitted from the source of vibration energy (e.g., the axle) to the 
suspended body (e.g., the vehicle structure). Therefore, the force generated by 
a semiactive damper is directly proportional to the relative velocity across the 
damper (just like a passive damper). Another class of dampers that is usually 
considered for vibration control is fully active dampers. Active dampers draw 

1203 


relatively substantial amounts of energy to produce forces that are not 
necessarily in direct relationship to the relative velocity across the damper. 

The virtues of active and semiactive dampers versus passive dampers 
have been addressed in many studies [3-10]. Using various analytical and 
experimental methods, these studies have concluded that in nearly all cases 
semiactive dampers reduce vibration transmission across the damper and 
better control the suspended (or sprung) body, in comparison to passive 
dampers. Further, they have shown that, for vehicle primary suspension 
systems, semiactive dampers can lower the vibration transmission nearly as 
much as fully active dampers; without the inherent cost and complications 
associated with active dampers. This has led to the prototype application, 
and production, of semiactive dampers for primary suspensions of a wide 
variety of vehicles, ranging from motorcycles, to passenger cars, to bus and 
trucks, and to military tanks, in favor of fully active systems. 

Although there is abundant research on the utility of semiactive 
dampers for improving vehicle ride and handling, their potential for reducing 
dynamic forces transmitted to the pavement remains relatively unexplored. 
This is because most suspension designers and researchers are mainly 
concerned with the role of suspension systems from the vehicle design 
perspective. Another perspective, however, is the effect of suspension 
systems on transmitting dynamic loads to the pavement. 


ROAD DAMAGE STUDIES 

Dynamic tire forces, that are heavily influenced by the suspension, are 
believed to be an important cause of road damage. Cole and Cebon [11] 
studied the design of a passive suspension that causes minimum road damage 
by reducing the tire force. They propose that there is a stronger correlation 
between the forth power of the tire force and road damage than the dynamic 
load coefficient (DLC) and road damage. 

A simple measure of road damage, introduced by Cebon m [12], is the 
aggregate fourth power force defined as 

Na 

Al=Y,Pjk k= 1,2,3 ..ns (1) 

where Pjk = force applied by tire j to point k along the wheel path, 
ria - number of axles on vehicle, and 

Us = number of points along the road.. 


1204 


DLC is a popular measurement frequently used to characterize dynamic 
loading and is defined as the root mean square (RMS) of the tire force divided 
by the mean tire force, which is typically the static weight of the vehicle. The 
equation takes the form; 


RMS Dynamic Tire Force 
Static Tire Force 

This study shows that minimum road damage, for a two degree-of-freedom 
model, is achieved by a passive system with a stiffness of about one fifth of 
current air suspensions and a damping of about twice that typically provided. 
In practice, however, reducing the suspension stiffiiess can severely limit the 
static load carrying capacity of the suspension and cause difficulties in vehicle 
operation. Further, higher damping can substantially increase vibration 
transmission to the body and worsen the ride. 

In another study by Cole and Cebon [13] a two-dimensional articulated 
vehicle simulation is validated with measurements from a test vehicle. The 
effect of modifications to a trailer suspension on dynamic tire forces are 
investigated. The RMS of dynamic loads generated by the trailer are predicted 
to decrease by 31 per cent, resulting in a predicted decrease in theoretical road 
damage of about 13 per cent. 

Yi and Hedrick compared the effect of continuous semiactive and 
active suspensions and their effect on road damage using the vehicle simulation 
software VESYM [14]. A control strategy based on the tire forces in a heavy 
truck model is used to show that active and semiactive control can potentially 
reduce pavement loading. They, however, mention that measuring the tire 
forces poses serious limitation in practice. 

The primary purpose of this paper is to extend past studies on 
semiactive suspension systems for reducing road damage. An alternative 
semiactive control policy, called "groundhook,” is developed such that it can 
be easily applied in practice, using existing hardware for semiactive 
suspensions. A simulation model representing a single primary suspension is 
used to illustrate the system effectiveness. The simulation results show that 
groundhook control can reduce the dynamic load coefficient and fourth power 
of tire force substantially, without any substantial increase in body 
acceleration. 


1205 


MATHEMATICAL FORMULATION 


We consider a model representing the dynamics of a single primary suspension 
in a heavy truck, as shown in Figure 1. 


Truck Mass 


Suspension 

Stiffness 

Suspension 
and Tire Mass 






Tire 

Stiffness 


Suspension 

Damping 


Road Input 


Figure 1. Primary Suspension Model 

This model has been widely used in the past for automobile applications, due 
to its effectiveness in analyzing various issues relating to suspensions. 
Although it does not include the interaction between the axles and the truck 
frame dynamics, the model still can serve as an effective first step in studying 
fundamental issues relating to truck suspensions. Follow up modeling and 
testing, using a full vehicle, is needed to make a more accurate assessment. 


The dynamic equations for the model in Figure 1 are: 


M^x^+C(x^-X 2 ) +K(x^~X 2 ) = 0 (3a) 

MjX, -C(ii -X2)-K{x^ -^ 2 ) + 0 

The variables Xi and X 2 represent the body and axle vertical displacement, 
respectively. The variable Xjn indicates road input, that is assumed to be a 
random input with a low-pass (0 - 25 Hz) filter. The amplitude for Xjn is 
adjusted such that it creates vehicle and suspension dynamics that resembles 
field measurements. Such a function has proven to sufficiently represent 
actual road input to the vehicle tires. 

Table 1 includes the model parameters, that are selected to represent a 
typical laden truck used in the U.S. The suspension is assumed to have a 
linear stiffness in its operating range. The damper characteristics are modeled 
as a non-linear function, as shown in Figure 2. 


1206 



Table 1. Model Parameters 


Description 

Symbol 

Value 

Body Mass 

Ml 

287 kg 

Axle Mass 

M2 

34 kg 

Suspension Stiffness 

K1 

196,142 N/m 

Tire Stifftiess 

Kt 

1,304,694 N/m 

Suspension Damping 

C 

See Table 2 


The bilinear function in Figure 2 represents the force-velocity 
characteristics of an actual truck damper. The parameters selected for both 
passive and semiactive dampers are shown in Table 2. These parameters are 
selected based on truck dampers commonly used in the U.S. Although we 
examined the effect of damper tuning on dynamic loading, if falls outside the 
scope of this paper. Instead, we concentrate here on comparing different 
semiactive dampers with a passive damper, using the baseline parameters 
shown in Table 2. The semiactive and passive damper characteristics used for 
this study are further shown in Figure 3. 



1207 






















Force Velocity Curve 



Figure 3. Passive and Semiactive Damper Characteristics 


SKYHOOK CONTROL POLICY 

As mentioned earlier, the development of semiactive dampers dates back to 
early 70’s when Kamopp and Crosby introduced the skyhook control policy. 
For the system shown in Figure 1, skyhook control implies 

X,(X] - ;C2) - 0 C = Con 
X](X^-X2)<0 C = Coff 

Where x, and represents the velocities of Mi (vehicle body) and M 2 (axle), 
respectively. The parameters Con and Coff represent the on- and off-state of 
the damper, respectively, as it is assumed that the damper has two damping 
levels. In practice, this is achieved by equipping the hydraulic damper with an 
orifice that can be driven by a solenoid. Closing the orifice increases damping 
level and achieves Con, whereas opening it gives Coff. 


1208 















Table 2. Damper Parameters 



Passive 

Semiactive On-State 

Semiactive Off-State 

n 

0.25 

0.35 

0.03 

m 

0.10 

0.15 

0.03 

Vbb 

0.254 m/sec 

0.254 m/sec 

0.254 m/sec 

B 

0.20 

0.30 

0.03 

S4 

0.10 

0.15 

0.03 

Vbj 

0.254 m/sec 

0.254 m/sec 

0.254 m/sec 


The switching between the two damper states, shown in Eq. (4), is 
arranged such that when the damper is opposing the motion of the sprung 
mass (vehicle body), it is on the on-state. This will dampen the vehicle body 
motion. When the damper is pushing into the body, it is switched to the off- 
state to lower the amount of force it adds to the body. Therefore, a semiactive 
damper combines the performance of a stiff damper at the resonance 
frequency, and a soft damper at the higher frequencies, as shovm in Figure 4. 



1209 



























Figure 5. Groundhook Damper Configurations: a) optimal groundhook 
damper configuration, b) semiactive groundhook damper 
configuration. 

This feature allows for a better control of the vehicle body, as has been 
discussed in numerous past studies. The skyhook control policy in Eq. (4), 
however, works such that it increases axle displacement, X 2 , (commonly called 
wheel hop). Because the tire dynamic loading can be defined as 


DL = KtX2 (5) 

The skyhook control actually increases dynamic loading. As mentioned earlier 
the development of skyhook policy was for improving ride comfort of the 
vehicle, without losing vehicle handling. Therefore, the dynamic loading of the 
tires was not a factor in the control development. 


GROUNDHOOK CONTROL POLICY 

To apply semiactive dampers to reducing tire dynamic loading, we propose an 
alternative control policy that can be implemented in practice using the same 
hardware needed for the skyhook policy. To control the wheel hop, this 
policy, called “groundhook,” implies: 


X, (;ci - X 2 ) < 0 C = Con 

(6a) 

x, (Xt - ^2) > 0 C = Coff 

(6b) 


As shovm in Figure 5, the above attempts to optimize the damping force on 
the axle, similar to placing a damper between the axle and a fictitious ground 
(thus, the name “groundhook”). The groundhook semiactive damper 
maximizes the damping level (i.e., C = Con) when the damper force is opposing 


1210 





the motion of the axle; otherwise, it minimizes the damping level (i.e., C = 
Coff). The damper hardware needed to implement groimdhook semiactive is 
exactly the same as 

the skyhook semiactive, except for the control policy programmed into the 
controller. 


SIMULATION RESULTS 

The model shown in Figure 1 is used to evaluate the benefits of groundhook 
dampers versus passive and skyhook dampers. A non-linear damper model 
was considered for the simulations, as discussed earlier. The road input was 
adjusted such that the dynamic parameters for the passive damper resembles 
actual field measurements. Five different measures were selected for 
comparing the dampers: 

• Dynamic Load Coefficient (DLC) 

• Fourth Power of the tire dynamic load 

• Sprung mass acceleration 

• Rattle Space (relative displacement across the suspension) 

• Axle Displacement, relative to the road 

Dynamic load coefficient and fourth power of tire force are measures 
of pavement dynamic loading and are commonly used for assessing road 
damage. They are both considered here because there is no clear consensus on 
which one is a better estimate of road damage. Axle displacement, relative to 
the road, indicates wheel hop and is directly related to DLC and tire force, 
therefore it is yet another measure of road damage. Sprung mass acceleration 
is a measure of ride comfort. Our experience, however, has shown that for 
trucks this may not be a reliable measure of the vibrations the driver feels in 
the truck. The relative displacement across the dampers relates to the rattle 
space, that is an important design parameter in suspension systems, 
particularly for cars. For each of the above measures, the data was evaluated 
in both time (Figures 6-7) and frequency domain (Figures 8 - 10). In time 
domain, the root mean square (RMS) and maximum of the data for a five- 
second simulation are compared. In frequency domain, the transfer function 
between each of the measures and road displacement is plotted vs. frequency. 
The frequency plots highlight the effect of each damper on the body and axle 
resonance frequency. 

Figures 6 and 7 show bar charts of root mean square (RMS) and 
maximum time data, respectively. In each case the data is normalized with 
respect to the performance of passive dampers commonly used in trucks. 
Therefore, values below line 1.0 can be interpreted as an improvement over the 
existing dampers. As Figures 6 and 7 show, groundhook dampers significantly 


1211 


improve pavement loading, particularly as related to the fourth power of tire 
force. Furthermore, the rattle space is improved slightly over passive 
dampers, indicating that groundhook dampers do not impose any additional 
burden on the suspension designers. 

One measure that has increased due to groundhook dampers is body 
acceleration. As mentioned earlier, in automobiles this measure is used as an 
indicator of ride comfort. In our past testing, however, we have found that for 
trucks it is a far less accurate measure of ride comfort. This is mainly due to 
the complex dynamics of the truck frame and the truck secondary suspension. 
A more accurate measure of ride comfort is acceleration at the B-Post (the 
post 



Tire Axle Body Rattle Space 

Dynamic Displacment Acceleration 

Force ^ 4 


Figure 6. RMS Time Data Normalized with respect to Passive Damper 



Tire Axle Body Rattle Space 

Dynamic Displacment Acceleration 

Force 4 


Figure 7. Max. Time Data Normalized with respect to Passive Damper 

1212 




behind the driver), which cannot be evaluated from the single suspension 
model considered here. Nonetheless, the body acceleration is included for the 
sake of completeness of data. 

The model shows that skyhook dampers actually increase the 
measures associated with pavement loading, while improving body 
acceleration. This agrees with the purpose of skyhook dampers that are 
designed solely for improving the compromise between ride comfort and 
vehicle handling. The improvement in ride comfort occurs at the expense of 
increased pavement loading. 



Figure 8. Transfer Function between Axle Displacement and Road Input 


Figures 8-10 show the frequency response of the system due to each damper. 
In each figure, the transfer fimction between one of the measures and input 
displacement is plotted vs. frequency. These plots highlight the impact of 
skyhook and groundhook on the body and wheel hop resonance, relative to 
existing passive dampers. The frequency plots indicate that the 


1213 




Transfer 


Frequency (Hz) 



groundhook dampers reduce axle displacement and fourth power of tire 
dynamic force at wheel hop frequency. At body resonance frequency, 
groundhook dampers do not offer any benefits over passive dampers. The 






frequency results for body acceleration and rattle space are similar to those 
discussed earlier for the time domain results. The frequency plots show that 
the skyhook dampers offer benefits over passive dampers at frequencies close 
to the body resonance frequencies. At the higher frequencies, associated with 
wheel hop, skyhook dampers result in a larger peak than either passive or 
groundhook dampers. This indicates that skyhook dampers are not suitable 
for reducing tire dynamic loading. 


CONCLUSIONS 

An alternative to skyhook control policy for semiactive dampers was 
developed. This policy, called “groundhook,” significantly improves both 
dynamic load coefficient (DLC), and fourth power of tire dynamic load, 
therefore holding a great promise for reducing road damage to heavy trucks. 
The dynamic model used for assessing the benefits of groundhook dampers 
represented a single suspension system. Although the results presented here 
show groundhook dampers can be effective in reducing tire dynamic loading 
and pavement damage, more complete models and road testing are necessary 
for more accurately assessing the benefits. 


REFERENCES 

1. Crosby, M. J., and Karnopp, D. C., "The Active Damper," The Shock 
and Vibration Bulletin 43, Naval Research Laboratory, Washington, D.C., 
1973. 

2. Karnopp, D. C., and Crosby, M. J., "System for Controlling the 
Transmission of Energy Between Spaced Members," U.S. Patent 
3,807,678, April 1974. 

3. Ahmadian, M. and Marjoram, R. H., “Effects of Passive and Semi-active 
Suspensions on Body and Wheelhop Control,” Journal of Commercial 
Vehicles, Vol. 98, 1989, pp. 596-604. 

4. Ahmadian, M. and Marjoram, R. H., “On the Development of a 
Simulation Model for Tractor Semitrailer Systems with Semiactive 
Suspensions,” Proceedings of the Special Joint Symposium on Advanced 
Technologies, 1989 ASME Winter Annual Meeting, San Francisco, 
California, December 1989 (DSC-Vol. 13). 

5. Hedrick, J. K., "Some Optimal Control Techniques Applicable to 
Suspension System Design," American Society of Mechanical Engineers, 
Publication No. 73-ICT-55, 1973. 


1215 


6. Hac, A., "Suspension Optimization of a 2-DOF Vehicle Model Using 
Stochastic Optimal Control Technique," Journal of Sound and Vibration, 
1985. 

7. Thompson, A. G., "Optimal and Suboptimal Linear Active Suspensions 
for Road Vehicles," Vehicle System Dynamics, Vol. 13,1984. 

8. Kamopp, D., Crosby, M. J., and Harwood, R. A., "Vibration Control 
Using Semiactive Force Generators," American Society of Mechanical 
Engineers, Journal of Engineering for Industry, May 1974, pp. 619-626. 

9. Krasnicki, E. J., "Comparison of Analytical and Experimental Results for 
a Semiactive Vibration Isolator," Shock and Vibration Bulletin, Vol. 50, 
September 1980. 

10. Chalasani, R.M., "Ride Performance Potential of Active Suspension 
Systems-Part 1: Simplified Analysis Based on a Quarter-Car Model," 
proceedings of 1986 ASME Winter Annual Meeting, Los Angeles, CA, 
December 1986. 

11. Cole, D. J. and Cebon, D., “Truck Suspension Design to Minimize Road 
Damage,” Proceedings of the Institution of Mechanical Engineers, Vol. 
210, D06894, 1996, pp. 95-107. 

12. Cebon, D., “Assessment of the Dynamic Forces Generated by Heavy 
Road Vehicles,” ARRB/FORS Symposium on Heavy Vehicle Suspension 
Characteristics, Canberra, Australia, 1987. 

13. Cole, D. J. and Cebon, D., “Modification of a Heavy Vehicle Suspension 
to Reduce Road Damage,” Proceedings of the Institution of Mechanical 
Engineers, Vol. 209, D03594, 1995. 

14. Yi, K. and Hedrick, J. K., “Active and Semi-Active Heavy Truck 
Suspensions to Reduce Pavement Damage,” SAE SP-802, paper 892486, 
1989. 


1216 



Active Vibration Control of Isotropic Plates Using 
Piezoelectric Actuators 


A. M. Sadri", J. R., Wright* and A. S. Cherry* 

The Manchester School of Engineering, Manchester M13 9PL, UK 

and 

R. J. Wynnes 

Sheffield Hallam University, School of Engineering, Sheffield, UK 


Abstract: Theoretical modelling of the vibration of plate components of a 
space structure excited by piezoelectric actuators is presented. The equations 
governing the dynamics of the plate, relating the strains in the piezoelectric 
elements to the strain induced in the system, are derived for isotropic plates 
using the Rayleigh-Ritz method. The developed model was used for a simply 
supported plate. The results show that the model can predict natural 
frequencies and mode shapes of the plate very accurately. The open loop 
frequency response of the plate when excited by the patch of piezoelectric 
material was also obtained. This model was used to predict the closed loop 
frequency response of the plate for active vibration control studies with 
suitable location of sensor-actuators. 


Introduction 

Vibration suppression of space structures is very important because 
they are lightly damped due to the material used and the absence of air 
damping. Thus the modes of the structure must be known very 
accurately in order to be affected by the controller while avoiding 
spillover. This problem increases the difficulty of predicting the 
behaviour of the structure and consequently it might cause unexpected 
on-orbit behaviour. 

These difficulties have motivated researchers to use the 
actuation strain concept. One of the mechanisms included in the 
actuation strain concept is the piezoelectric effect whereby the strain 
induced through a piezoelectric actuator is used to control the 


Research Student, Dynamics &: Control Research Group. 
^ Professor, Dynamics & Control Research Group, 

^ Former Lecturer, Dynamics & Control Research Group. 
^ Professor of Mechanical and Control Engineering. 


1217 


deformation of the structure [1]. It can be envisaged that using this 
concept in conjimction with control algorithms can enhance the ability 
to suppress modes of vibration of flexible structures. 

Theoretical and experimental results of the control of a flexible 
ribbed antenna using piezoelectric materials has been investigated in 
[2]. An active vibration damper for a cantilever beam using a 
piezoelectric polymer has been designed in [3]. In this study, 
Lyapunov's second or direct method for distributed-parameter systems 
was used to design control algorithms and the ability of the algorithms 
was verified experimentally. These works have clearly shown the 
ability of piezoelectric actuators for vibration suppression. However, 
they have been limited to one dimensional systems. Obviously, there is 
a need to understand the behaviour of piezoelectric actuators in two 
dimensional systems such as plates. 

Vibration excitation of a thin plate by patches of piezoelectric 
material has been investigated in [4]. Their work was basically an 
extension of the one dimensional theory derived in [1] to show the 
potential of piezoelectric actuators in two dimensions. In their studies, 
it was assumed that the piezoelectric actuator doesn't significantly 
change the inertia, mass or effective stiffness of the plate. This 
assumption is not guaranteed due to the size, weight and stiffness of 
the actuator. Based on this assumption, their model can not predict the 
natural frequencies of the plate accurately after bonding piezoelectric 
actuators. Therefore, it is essential to have a more general model of a 
plate and bonded piezoelectric actuators with various boundary 
conditions. The model should be able to predict frequency responses 
because this is fundamental to the understanding of the behaviour of 
the system for control design purposes. It is the objective of the current 
study to develop such a modelling capability. 

Previous work [5, 6, 7] has concentrated on the modelling and 
control of a cantilever beam. The method used involved bonding 
piezoelectric material to a stiff constraining layer, which was bonded to 
the beam by a thin viscoelastic layer in order to obtain both active and 
passive damping. Then a Rayleigh-Ritz model was developed and 
used to derive a linearized control model so as to study different 
control strategies. In the work described in this paper, the method has 
been extended to the more complex plate problem. The paper 
introduces a modelling approach based on the Rayleigh-Ritz assumed 
mode shape method to predict the behaviour of a thin plate excited by 
a patch of piezoelectric material bonded to the surface of the plate. The 
model includes the added inertia and stiffness of the actuator and has 
been used to predict the frequency response of the plate. Suggestions 
for future work are also included. 


1218 


Theoretical Modelling 

In developing the Rayleigh-Ritz model of a plate excited by a 
patch of piezoelectric material bonded to the surface of the plate, a 
number of assumptions must be made. The patch of piezoelectric 
material is assumed to be perfectly bonded to the surface of the plate. 
The magnitude of the strains induced by the piezoelectric element is a 
linear fimction of the applied voltage that can be expressed by 


e:=E: 


( 1 ) 


Here is the piezoelectric strain constant, is the piezoelectric layer 

thickness and V 33 is the applied voltage. The index 31 shows that the 
induced strain in the ' 1 ' direction is perpendicular to the direction of 
poling '3' and hence the applied field. The piezoelectric element 
thickness is assumed to be small compared to the plate thickness. The 
displacements of the plate middle surface are assumed to be normal to 
it due to the bending affects. 

Figure 1 shows the configuration of the bonded piezoelectric 
material relative to the surface of the plate. 


w 



Figure 1. Configuration of the bonded piezoelectric actuator on the surface of the 
plate. 

In figure 1, 4 and 4 are the dimensions of the plate, x,, X 2 , y, and y^ 
are the boundaries of the piezoelectric element and w, v and w are the 
displacements in the x , y and z direction, respectively. 

To derive the equations of motion of the plate based on the 
Rayleigh-Ritz method, both the strain energy U and kinetic energy T of 
the plate and the piezoelectric element must be determined. The strain 


1219 



and kinetic energy result from the deformation produced by the 
applied strain which is induced by exciting the piezoelectric element. 
The deformations can be expressed by the combination of the midplane 
displacement and the deformations resulting from the bending of the 
plate. 


Strain Energy 

The strain energy of the plate and piezoelectric material can be 
calculated by 

U = +T;«yY,<y)dVp +T^T„)dVp. (2) 

where 8 is the inplane direct strain, a is the inplane direct stress, t is 
the inplane shear stress and y is shear strain. dV shows volume 
differential and indices p and pe refer to the plate and piezoelectric 
actuator, respectively. The strains 8^ , 8^ and y can be shown to be 

For the Plate: 

du 

dv d (3) 

_du dv d^w 

^ dy dx dxdy 

For the Piezoelectric actuator: 


Ev = 


du 

a 

^~'dx~ 

^ a?* 

Sv 

a 



du av 
« ■ _ 1 ^ 

dy dx 




'"33=S,-H'^33 


a 


dxdy ^ 


( 4 ) 


where _ refers to the strains due to the deformation. The stresses , 
Gy , y can be expressed as 


1220 


For the Plate: 



E„ 

1 


0 

‘e. ' 




1 

0 




y.vy. 


0 

0 


r.y_ 





2 . 



( 5 ) 


For the Piezoelectric actuator: 




' 




0 ^. 

II 

1 


0 

Sx-H ^33 



1 

0 


r.y_ 

0 

0 

l-v 

2 . 

. Is 


( 6 ) 


where E is Young modulus and v is Poisson's ratio for the assumed 
isotropic material. Substituting equations 3, 4, 5 and 6 into 2 yields the 
strain energy of the plate and piezoelectric actuator. 


^ S, + sj + 1 (1 - Dp) dVp 
- '"''33)' + 2Dpp( e, - HV33){ E, - M.V33 )^ + ^ 1 ciVpp. 

•'Vp, ^ _ u — - tC 


( 7 ) 


Kinetic Energy 

To obtain the kinetic energy, the velocity components in x, y and 
z directions are needed. The velocity components can be calculated by 
differentiating the displacement components which are 

dw 

u— u— z— 
dx 

dw 

( 8 ) 

dy 

w=w. 

Differentiating equations 8 yields 


1221 


dw 

u= u- z — 
ax 

dw 

v= V-z— 

dy 

w 


( 9 ) 


where u , y and w are the velocity components in the x, y and z 
directions respectively. Using these velocity components, the kinetic 
energies of the plate and piezoelectric actuator are obtained as 



dw 




P rw^ + (u-z^f+ (v-z^^]dV^ 


9x 


ay' 



( 10 ) 


where p is the mass density. 


Equation of Motion 

The static or dynamic response of the plate excited by the 
piezoelectric actuator can be calculated by substituting the strain and 
kinetic energy into Lagrange's equation 


d dT dT ^ dU 
dt dq. dq,"^ dq. 


( 11 ) 


where q^ represents the ith generalised coordinate and is the ith 
generalised force. As there are no external forces (the force applied by 
5 \e piezoelectric element is included as an applied strain) or gyroscopic 
terms and there is no added damping, Lagrange's equation reduces to : 


dt dq, ^ dq, 


( 12 ) 


Now the equation of motion can be obtained by using the expression 
obtained for the strain and kinetic energy, and the assumed shape 
functions for flexural and longitudinal motion 


1222 


u(x,y,t)~ 

v(x,y,t)={i,(x,y)y 

w(x, y, t) = {(|)(j:, y)}^ {?(?)}. 


(13) 


Here y , ^ and ([) are the assumed displacement shape and h , f and g 
are generalised coordinate of the plate response m x, y and z 
directions. Using the shape functions expressed in equations 13, 
substituting equations 7 and 10 into equation 12, and including 
Rayleigh damping yields the equation of motion of the plate in the 
form 




where M, C and K are mass, damping and stiffness matrices and P is 
the voltage-to-force transformation vector. Vector q represents the 
plate response modal amplitudes and V is the applied voltage. 

State-Space Equations 

A model of a structure found via finite element or Rayliegh-Ritz 
methods results in second-order differential equations of the form 


[M]{9}+[C]{?}+[if]{?}={P}V (15) 

Choosing state variables x^_ = q and = i,, equation 1 may be 

reduced to a state-space representation as follows : 

q = x =X2 

/ / / (16) 
q = X2= ~M-^Kq - M~'Cq + M~'PV. 


Equations 15 can then be rewritten as 




r 0 /I 



0 


1 

1 

1 

1 

A. 

+ 


^2. 


.^2_ 




[,] = [/ 0 ] 


( 17 ) 


where [ ], { } are ignored. It should be noted that the vector q must be 
multiplied by the shape fimctions to produce the actual displacement. 


1223 


Results 

The model was used to investigate the response of a simply 
supported plate. In order to maintain symmetry of the geometric 
structure a piezoelectric actuator is assumed to be bonded to both the 
top and bottom surfaces of the plate. So The symmetry of the elements 
causes no extension of the plate midplane and the plate deforms in 
pure bending. In this case the shape functions are assumed to be : 

\{/(x,y) = 0 

^(x,y) = 0 (18) 

41 (X, y) = jinf x)sin( ^ y) 

A- 

where m and n are the number of half waves in the x and y directions. 

The properties of the plate are given in Table 1 and its 
dimensions are 4 = Q38m , 4 , = Q30/w and 4=i.5876m/K. Tables 2 and 3 
show the natural frequencies of the bare plate obtained by the thin 
plate theory and the RR model, respectively. Since the shape functions 
used in this example express the exact shape of the simply supported 
plate, the natural frequencies included in Tables 2 and 3 are very close. 

In order to excite the plate, a piezoelectric actuator with 
configuration x, = 0.32 Xj = 0.36 m, = 0.04 OT and = 026 m is used above 
and below. The natural frequencies of the plate after bonding the 
piezoelectric actuator to the surface are given in Table 4. The results 
show an increase in natural frequencies, showing that the added 
stifness is more important than the added inertia. 

Table 1: Properties of the plate 

207 7870 .292 


Table 2 : Plate natural frequencies (rad / s), Thin Plate Theory 



1 

2 

3 

4 

1 

437.5 

1246.0 

2593.5 

4480.0 

2 

941.4 

1749.9 

3097.4 

4983.9 

3 

1781.2 

2589.7 

3937.2 

5823.7 

4 

2957.0 

3765.5 

5113.0 

6999.5 


1224 


Table 3 : Plate natural frequencies (rad / s), RR Model 


/n 

1 

J 

2 

3 

4 

1 

437.5 

1245.9 

2593.2 

4479.1 

2 

941.3 

1749.7 

3097.0 

4982.8 

3 

1781.1 

2589.4 

3936.6 

5822.3 

4 

2956.7 

3764.9 

5111.9 

6997.4 


Table 4 : Plate-Piezo natural frequencies (rad / s) 


/n 

1 

2 

3 

4 

1 

444.0 

1257.1 

2611.3 

4502.8 

2 

957.2 

1775.6 

3182.1 

5076.7 

3 

1854.6 

2642.5 

4072.3 

6029.6 

4 

3076.6 

3933.8 

5224.6 

7277.2 


Initially, the piezoelectric actuator was excited by a constant DC 
voltage. The result of this action is shown in figure 2 which shows the 
dominant out of plane displacement around the location of the 
piezoelectric actuator bonded to the surface of the plate. To show the 
modes of vibration, the piezoelectric actuator was excited by a voltage 
with frequencies near to the natural frequencies of modes ( 2 , 2 ) and 
(1,3). The response in figures 3,4 show that the piezoelectric actuator 
excited both of these modes. 


Displacement Distribution (X-Y) 



0 0 


Figure 2. Static Displacement 


1225 






Displacement Distribution (X-Y) 



0 0 


Figure 3. Vibration of the plate, mode (2,2) 


Displacement Distribution (X-Y) 



0 0 


Figure 4. Vibration of the plate, mode (3,1) 

The frequency response of tihe plate at the centre was obtained 
by exciting the piezoelectric actuator at a range of frequencies between 
0 and 4000 rad/s. Figure 5 shows the frequency response of the plate 
at its centre. The frequency response of the plate at y = 0.5L^ along the 
x-direction is shown in figure 6. It can be seen that the amplitude of 
vibration of some modes are very high compared to that of the other 
modes. Special attention must be given for the suppression of vibration 
of these modes. 


1226 





Frequency Response of the Plate 



Figure 5. Frequency response of the plate at the centre 


Frequency Response of the Plate 



4000 


Figure 6. Frequency response of the plate along x-direction 

The results show that it is possible to predict the frequency 
response of a plate when it is excited by a patch of piezoelectric 
material. Consequently, a sensor model can be also added to the model 
and a signal proportional to velocity fed back to the piezoelectric 
patch. As a result, the closed loop frequency response of the plate can 
be obtained theoretically which is very important for active vibration 
control studies. This also permits the investigation of the optimal 


1227 












location of the actuators and the study of control algorithms for the 
best possible vibration suppression before using any costly 
experimental equipment. 

For this purpose, two patches of piezoelectric (lOcmxScm and 
5 cmx4cm), whose specifications are listed in table 5, were bonded to 
the surface of the plate in different locations and then the plate was 
excited by a point force marked by "D" in the figure 7. In figure 7 the 
dash lines are showing the nodal lines of a simply supported plate up 
to mode (3,3). 


Table 5: Properties of the actuator 
(mm) EJxlO''^N/m')pJkg/m') dJxlO'^'mfv) ^ 

.2 6.25 7700 -180 .3 


An actuator is most effective for control of a particular mode if 
the sign of the strain due to the modal deflection shape is the same 
over die entire actuator. Consequently, as can be seen from figure 7, 
the actuators are placed between the nodal lines and at the points of 
maximum curvature in order to obtain good damping effect on the 
modes of interest. Then two accelerometers were located at the center 
of the location of the actuators, marked by "S" in figure 7, in order to 
have collocated sensor-actuators. The signals obtained by the 
accelerometers are integrated and fed back to the actuators separately. 
Therefore rate feedback was used in this configuration. This leads to 
the feedback control law 


V = kq 


(19) 


where k is an amplification factor or feedback gain. Substituting 
equation 19 into equation 17 the closed-loop state-space representation 
of the system can then be obtained as 




0 

-M-‘K 


_J 

I — 

1 

M-^(C-kP)_ 



[^]=[/ 0 ] 


( 20 ) 


1228 


Then the effects of the actuators on vibration suppression were 
investigated. At first, only the actuator near to the center of the plate 
was used to suppress the vibration. The effect of this is shown in figure 
8 . As can be seen, damping in some modes are improved and some 
modes are untouched. 

The second actuator was added to the model to see its effect on 
modes of vibration. 



Figure 7. Plate with Bonded Piezoelectric Actuators 


Frequency Response of the Plate 



Figure 8. Open and Closed loop Frequency Response of the plate 

solid line; open loop, dash line : closed loop 


1229 






















Frequency Response of the Plate 


-50 



_2qqI-1-1-1-1-1-1——-1-1 

0 500 1000 1500 2000 2500 3000 3500 4000 

Frequency (rad/s) 

Figure 9. Open and Closed Loop Frequency Response of The Plate 
solid line: open loop, dash line : closed loop 


The open and closed loop frequency response of the plate when excited 
by the point force and controlled by two actuators is shown in figure 9. 
As can be seen, significant vibration suppression was obtained in both 
lower and higher modes. Also, it shows that the place of actuators was 
successfully chosen. This analysis showed that obtaining reasonable 
but not necessarily optimal placement of actuators in structures is very 
important in order to obtain a high level of damping in the modes of 
interest. Obviously, bonding more than one piezoelectric actuator in 
suitable locations helps to successfully suppress vibration of the plate. 

Conclusions 

A model of an active structure is fundamental to the design of 
control strategies. It can be used to analyse the system and investigate 
optimal control strategies without using costly experimental 
equipment. 

A Rayleigh-Ritz model has been developed to analyse the behaviour 
of a thin plate excited by a patch of piezoelectric material. The model 
has been used for a simply supported plate. It has been shown that the 
model can predict natural frequencies of the plate alone very 
accurately. The obtained mode shapes also correspond to the actual 
mode shapes. The frequency response of the plate can be obtained to 
show the suitability of the model for control design studies. This study 
allowed the behaviour of the system in open and closed loop form for 
active vibration control purposes to be investigated. Two piezoelectric 
actuators were used to investigate their effectiveness on vibration 


1230 
















suppression of the plate. The analysis showed that the location of two 
actuators was very important to increase the level of damping in both 
lower and higher frequency modes. The future work will be to extend 
the model to analyse a plate with more than two patches of 
piezoelectric material with optimal configuration, obtained by 
controllability theory, and independent controller for vibration 
suppression, and experimental verification of the theoretical analysis. 


References 

1. Crawley, E. F. and de Luis, J., Use of Piezoelectric Actuators as 
Elements of Intelligent Structures, AlAA Journal, Vol. 25, No. 10, 
1987, p. 1373. 

2. Dosch, J., Leo, D. and Inman, D., Modelling and control for 
Vibration Suppression of a Flexible Active structure, AIAA Journal 
of Guidance, Control and Dynamics, Vol. 18, No. 2,1995, p.340. 

3. Bailey, T. and Hubbard J. E. Jr., Distributed Piezoelectric Polymer 
Active Vibration Control of a Cantilever Beam, AIAA Journal of 
Guidance, Control and Dynamics, Vol. 8, No. 4,1985, p.605. 

4. Dimitriadis, E. K., Fuller, C. R., Rogers C. A., Piezoelectric 
Actuators for Distributed Vibration Excitation of Thin Plates, 
Journal of Vibration and Acoustics, Vol. 113, No. 1,1991, p. 100. 

5. Azvine, B., Tomlinson, G. R. and Wynne, R. J., Use of Active 
Constrained Layer Damping for Controlling Resonant, Journal of 
Smart Materials and Structures, No. 4,1995. 

6 . Rongong, J. A., Wright, J. R., Wynne, R. J. and Tomlinson, G. R., 
Modelling of a Hybrid Constrained Layer/Piezoceramic Approach 
to Active Damping, Journal of Vibration and Acoustics, To appear. 

7. Sadri, A. M., Wynne, R. J. and Cherry, A. S., Modelling and 
Control of Active Damping for Vibration Suppression, UKACC 
International Conference on Control' 96, 2-5 September 1996. 

8 . Bathe, K,, Finite Element Procedures in Engineering Analysis, Prentice- 
Hall, Inc., 1982. 

9. Blevins, R. D., Formulas for Natural Frequency and Mode Shapes, Van 
Nostrand Remhold, NY 1979. 

10. Thomson, W. T., Theory of Vibration with Applications, Prentice-Hall, 
Inc., 1988. 


1231 





Active control of sound transmission into a rectangular 
enclosure using both structural and acoustic actuators 


S.M. Kim and MJ. Brennan 

ISVR, University of Southampton, Highfieid, Southampton, S017 IBJ, UK 


ABSTRACT 

This paper presents an analytical investigation into the active control of sound 
transmission in a ‘weakly coupled’ structural-acoustic system. The system 
under consideration is a rectangular enclosure having one flexible plate 
through which external noise is transmitted. Three active control systems 
classified by the type of actuators are discussed. They are; i) a single force 
actuator, ii) a single acoustic piston source, and iii) simultaneous use of both 
the force actuator and the acoustic piston source. For all three control systems 
the acoustic potential energy inside the enclosure is adopted as the cost 
function to minimise, and perfect knowledge of the acoustic field is assumed. 
The results obtained demonstrate that a single point force actuator is effective 
in controlling well separated plate-controlled modes, whereas, a single 
acoustic piston source is effective in controlling well separated cavity- 
controlled modes provided the discrete actuators are properly located. Using 
the hybrid approach with both structural and acoustic actuators, improved 
control effects on the plate vibration together with a further reduction in 
transmitted noise and reduced control effort can be achieved. Because the 
acoustic behaviour is governed by both plate and cavity-controlled modes in a 
‘weakly coupled’ structural-acoustic system, the hybrid approach is desirable 
in this system. 


1. INTRODUCTION 

Analytical studies of vibro-acoustic systems have been conducted by many 
investigators to achieve physical insight so that effective active control 
systems can be designed. It is well established that a single point force actuator 
and a single acoustic piston source can be used to control well separated 
vibration modes in structures and well separated acoustic modes in cavities, 
respectively, provided that the actuators are positioned to excite these 
modes [1,2]. Active control is also applied to structural-acoustic coupled 
systems for example, the control of sound radiation from a piate[3-6] and the 
sound transmission into a rectangular enclosure[7-8]. Meirovitch and 


1233 


Thangjitham[6], who discussed the active control of sound radiation from a 
plate, concluded that more control actuators resulted in better control effects. 
Pan et al[Z] used a point force actuator to control sound transmission into an 
enclosure, and discussed the control mechanism in terms of plate and cavity- 
controlled modes. 

This paper is concerned with the active control of sound transmission into a 
‘weakly coupled’ structural-acoustic system using both structural and acoustic 
actuators. After a general formulation of active control theory for structural- 
acoustic coupled systems, it is applied to a rectangular enclosure having one 
flexible plate through which external noise is transmitted. Three active control 
systems classified by the type of actuators are compared using computer 
simulations. They are; i) a single force actuator, ii) a single acoustic piston 
source, and iii) simultaneous use of both the force actuator and the acoustic 
piston source. For all three control systems the acoustic potential energy inside 
the enclosure is adopted as the cost function to minimise, and perfect 
knowledge of the acoustic field is assumed. The effects of each system are 
discussed and compared, and a desirable control system is suggested. 


2. THEORY 

2.1 Assumptions and co-ordinate systems 

Consider an arbitrary shaped enclosure surrounded by a flexible structure and a 
acoustically rigid wall as shown in Figure 1. A plane wave is assumed to be 
incident on the flexible structure, and wave interference outside the enclosure 
between the incident and radiated waves by structural vibration is neglected. 
Three separate sets of co-ordinates systems are used; Co-ordinate x is used for 
the acoustic field in the cavity, co-ordinate y is used for the vibration of the 
structure, and co-ordinate r is used for the sound field outside the enclosure. 
The cavity acoustic field and the flexible structure are governed by the linear 
Helmholtz equation and the isotropic thin plate theory[9], respectively. The 
sign of the force distribution function and normal vibration velocity are set to 
be positive when they direct inward to the cavity so that the structural 
contribution to acoustic pressure has the same sign as the acoustic source 
contribution to acoustic pressure. 

Weak coupling rather than full coupling is assumed between the structural 
vibration system and the cavity acoustic system. Thus, the acoustic reaction 
force on the strucural vibration under structural excitation and the structural 
induced source effect on the cavity acoustic field under acoustic excitation is 
neglected. This assumption is generally accepted when the enclosure consists 
of a heavy structure and a big volume cavity. It is also assumed that the 
coupled response of the system can be described by finites summations of the 


1234 


uncoupled acoustic and structural modes. The uncoupled modes are the rigid- 
walled acoustic modes of the cavity and the in vacuo structural modes of the 
structure. The acoustic pressure and structural vibration velocity normal to the 
vibrating surface are chosen to represent the responses of the coupled system. 

2.2 Structural-acoustic coupled response 

The acoustic potential energy in the cavity is adopted as the cost function for 
the global sound control, which is given by [2] 

where, and Co respectively denote the density and the speed of sound in air, 
and /?(x,ft)) is the sound pressure inside the enclosure. 

The vibrational kinetic energy of the flexible structure, which will be used to 
judge the control effect on structural vibration, is given by[l] 

where, p., is the density of the plate material, h is the thickness of the plate. 

If the acoustic pressure and the structural vibration are assumed to be 
described by a summation of N and M modes, respectively, then the acoustic 
pressure at position x inside the enclosure and the structural vibration velocity 
at position y are given by 

N 

P(X,(B) = 

n=I 

M 

ni=l 

where, the N length column vectors ^ and a consist of the array of uncoupled 
acoustic mode shape functions and the complex amplitude of the 

acoustic pressure modes a^^(co) respectively. Likewise the M length column 

vectors O and b consist of the array of uncoupled vibration mode shape 
functions 0n,(y) and the complex amplitude of the vibration velocity modes 

respectively. 

The mode shape functions \f/^{x) and (l)Jy) satisfy the orthogonal property 
in each uncoupled system, and can be normalised as follows. 

V = lwlMdV (5) 

S,=lfyy)dS ( 6 ) 


1235 


where, V and S/ are the volume of the enclosure and the area of the flexible 
structure, respectively. Since mode shape functions are normalised as given by 
(Eq. 5), the acoustic potential energy can be written as 

£ =_L^a“a (7) 

' 4p„c^ 

Similarly from (Eq. 2) and (Eq. 6), the vibrational kinetic energy can be 
written as 

£ =££^b«b ( 8 ) 

2 

Where superscript H denotes the Hermitian transpose. 

For the global control of sound transmission, it is required to have knowledge 
of the complex amplitude of acoustic pressure vector a for various excitations. 

The complex amplitude of the n-th acoustic mode under structural and 
acoustic excitation is given by[9,10] 

^ 1 . (^) = 4 (^)(^ (x)‘y(x, co)dv +i//'„ (yMy, j (9) 

where, s(x,Ci)) denotes the acoustic source strength density function in the 
cavity volume V”, and w(y,6)) denotes the normal velocity of the surrounding 
flexible structure on surface 5/-. The two integrals inside the brackets represent 
the nth acoustic modal source strength contributed from s(x,Q)) and u(y,co ), 
respectively. The acoustic mode resonance term An(fi)) is given by 

A,.(co) = - -- ( 10 ) 

o}--Q)-+j2C„<a„(o 

where a)„and C„ are the natural frequency and damping ratio of the nth 
acoustic mode, respectively. 

Substituting (Eq. 4) into (Eq. 9) and introducing the modal source strength 
q,: = j ¥n X. 0))dV , then we get 


2 / M 

a „(®) = A,(®) (®) + SO',,™ • b,„((0) 

y V Hi=i 


where, C„,,„ represents the geometric coupling relationship between the 
uncoupled structural and acoustic mode shape functions on the surface of the 
vibrating structure Sf and is given by[l 1] 

c,,,„ = lvJy)<t>Jy’0>)ds ( 12 ) 

If we use L independent acoustic control sources, can be written as 

/=! \l /=! 


1236 


where =- j W^ control source strength q,^i{co) 

having an area of Scjj is defined at Xc,i. 

Thus, the complex amplitude of acoustic modal pressure vector a can be 
expressed as 

a = Z.(D,q,+Cb) (14) 

2 

whereZa= -^A. 

The matrix A is a (NxN) diagonal matrix in which each (n,n) diagonal term 
consists of A„, the (NxM) matrix C is the structural-acoustic mode shape 
coupling matrix, the (NxL) matrix Dq determines coupling between the L 
acoustic source locations and the N acoustic modes, the L length vector (jc is 
the complex strength vector of acoustic control sources, and b is the complex 
vibrational modal amplitude vector. The {NxN) diagonal matrix Za can be 
defined as the uncoupled acoustic modal impedance matrix which determines 
the relationship between the acoustic source excitation and the resultant 
acoustic pressure in modal co-ordinates of the uncoupled acoustic system. 
Generally the impedance matrix is symmetric but non-diagonal in physical co¬ 
ordinates, however the uncoupled modal impedance matrix is diagonal 
because of the orthogonal property of uncoupled modes in modal co-ordinates. 


Since the flexible structure in Figure 1 is assumed to be governed by the 
isotropic thin plate theory, the complex vibrational velocity amplitude of the 
mth mode can be expressed as[10] 

= + (15) 

p,hSf ^ 

where, again p.v is the density of the plate material, h is the thickness of the 
plate, is the area of flexible structure. Inside the integral /(y,ty), p""\'^.(0 ), 
and p‘"'(y,ty) denote the force distribution function, and the exterior and 
interior sides of acoustic pressure distribution on the surface 5/, respectively. 
Because of the sign convention used, there is a minus sign in front of 
(y, (O ). The structural mode resonance term Bm((0) can be expressed as 




-(0^+j2^„^0),„co 


where co^ and Cm natural frequency and the damping ratio of m-th 


mode, respectively. Substituting (Eq. 3) into (Eq. 15), then we get 

b,niO})=——B,„ico)\ g,,,„ico)+ g^,„i(o)-'£cl„-a,(co) (17) 

p,hSf \ „=i ; 


1237 


where, L = L 

• JSj- •'■i/ 

and Cl^ = C,„.. ■ 

If we use K independent point force actuators, the m-th mode generalised force 
due to control forces, gc.m, can be written by 

= XI <P,n(y)S(y-ycjc)dsf,j,ico) = X^/.».t/<.-.t(®) 

k=\ ^ k=] 

( 18 ) 

where 0,„(y)5(y-y.,t)rfS, and the k-th control point force l,{co) 

JSj 

is located at y^-it. 

Thus the modal vibrational amplitude vector b can be expressed as 

b = Y,(g,+Dff,-C’'a) (19) 

where Ys = —^— B. 

P..hSjr 

The matrix B is a (MxM) diagonal matrix in which each (m,m) diagonal term 
consists of B,„, is the transpose matrix of C, the (NxK) matrix Df 
determines coupling between the K point force locations and the M structural 
modes, gp is the generalised modal force vector due to the primary plane wave 
excitation, the K length vector fc is the complex vector of structural control 
point forces, and a is the complex acoustic modal amplitude vector. The 
(MxM) diagonal matrix Ys can be defined as the uncoupled structural modal 
mobility matrix which determines the relationship between structural 
excitation and the resultant structural velocity response in modal co-ordinates 
of the uncoupled structural system. As with the uncoupled acoustic impedance 
matrix Za, note that Ys is a diagonal matrix. 


From (Eq. 14) and (Eq. 19), we get 



a = (l + Z,CY,C’^)''z,(D,q,+CY,g,+CWc) 

(20) 


b = (I + Y,C’'Z,C)'' Y,(gp + Dff, - C^Z.D^q,) 

(21) 

Since weakly coupling is assumed i.e. = 0 and Y^C^Z^C = 0 , 

then 

we get 

a = Z,(D,q,+CY,g,+CY,D,f.) 

(22) 


b = X(g,+D,f,-C%D,q,) 

(23) 


Although the formulation developed above covers fully coupled systems, weak 
coupling is assumed hereafter for the convenience of analysis. 


1238 



In order to minimise the sound transmission into the cavity, two kinds of 
actuators are used: a single point force actuator for controlling the structural 
vibration of the plate and a single rectangular type acoustic piston source for 
controlling the cavity acoustic pressure. The rectangular piston source is 
centred at (1.85,0.15,0) with the area of 0.15m by 0.15m, This location was 
chosen because the sound pressure of each mode in a rectangular cavity is a 
maximum at the corners, and thus the control source is placed away from the 
acoustic nodal planes [2]. For a similar reason, the point force actuator is 
located at (9/20Li, L 2 / 2 ) on the plate, at which there are no nodal lines within 
the frequency range of interest. Table 2 shows the natural frequencies of each 
uncoupled systems and their geometric mode shape coupling coefficients 
which are normalised by their maximum value. Some of natural frequencies 
which are not excited by the given incident angie((p = 0°) were omitted. The 
(m/, m 2 ) and («/, 112 , ns) indicate the indices of the m-th plate mode and the n\h 
cavity mode, and corresponding the uncoupled natural frequencies of the plate 
and the cavity are listed. A total 15 structural and 10 acoustic modes were used 
for the analysis under 300 Hz, and no significant difference was noticed in 
simulations with more modes. 


3.2 Active minimisation of the acoustic potential energy 

This section considers an analytical investigation into the active control of the 
sound transmission into the rectangular enclosure in Figure 2. Three active 
control strategies classified by the type of actuators are considered. They are; 
i) a single force actuator, ii) a single acoustic piston source, and iii) 
simultaneous use of both the force actuator and the acoustic piston source. 
Although the formulation developed in this paper is not restricted to a single 
actuator, each single actuator was used to simplify problems so that the control 
mechanisms could be understood and effective guidelines for practical 
implementation could be established. 

3.2.1 control using a single force actuator 

A point force actuator indicated in Figure 2 is used as a structural actuator and 
the optimal control strength of the point force actuator can be calculated using 
(Eq. 26). Figure 3(a) shows the acoustic potential energy of the cavity with 
and without the control force. To show how this control system affects the 
vibration of the plate, the vibration kinetic energy of the plate obtained from 
(Eq. 8) is also plotted in Figure 3(b). On each graph, natural frequencies of the 
plate and the cavity are marked and ‘o’ at the frequencies, respectively. It 
can be seen that the acoustic response of uncontrolled state has peaks at both 


1239 


plate and cavity resonances, and the vibration response of uncontrolled state is 
governed by the plate resonances only because of ‘weak coupling’. 

Examining Figure 3(a,b) it can be seen that at the 1st, 2nd, 4th, and 5th plate 
modes corresponding to 52 Hz, 64 Hz, 115 Hz, and 154 Hz, respectively there 
is a large reduction of the acoustic potential energy. This is because the sound 
field at these frequencies is governed by the plate vibration modes, and a 
single structural actuator is able to control the corresponding vibration mode to 
minimise sound transmission. 

The structural actuator reduces sound at cavity-controlled modes as well( 
especially the 2nd and 3rd cavity modes corresponding respectively to 85 Hz 
and 170 Hz), however it has to increase plate vibration significantly. It shows 
that minimisation of the acoustic potential energy does not always bring the 
reduction of structural vibration, and vice versa. Since a cavity-controlled 
mode is generally well coupled with several structural modes, a single 
structural actuator is not able to deal with several vibration modes because of 
'control spillover\A\. This is the reason why a single acoustic piston source 
used in the next section was introduced. However, it is clear that a single 
point force actuator is effective in controlling a well separated plate-controlled 
mode provided the actuator is not located close to the nodal line. 


3.2.2 control using a single piston force source 

A single acoustic piston source indicated in Figure 2 is used for controlling 
the acoustic sound field directly. The optimal control source strength of the 
acoustic piston source can be determined using (Eq. 26). Figure 4 shows the 
acoustic potential energy of the cavity and the vibrational kinetic energy of the 
plate with and without the control actuators. 

Since a plate-controlled mode is generally coupled with several cavity modes, 
the control effect of the acoustic source is not effective at plate-controlled 
modes (e.g. 52 Hz, 64 Hz, 115 Hz etc.). Whereas, it is more able to reduce 
transmitted sound at the cavity-controlled modes ( e.g. 0 Hz, 85 Hz, 170 Hz 
and 189 Hz) than the structural actuator. As can be noticed from (Eq. 22), the 
external incident wave and force excitation have the same sound transmission 
mechanism, which is vibrating the plate and transmitting sound through the 
geometric mode shape coupling matrix C. Thus, it can be said that the 
structural actuator is generally effective in controlling sound transmission. At 
cavity controlled modes, however, several vibration modes are coupled with 
an acoustic mode. It means that a single acoustic source is more effective than 
a single structural actuator since a single actuator is generally able to control 
only one mode. From the results, it is clear that a single acoustic piston source 
is effective in controlling well separated cavity-controlled modes. It is 
interesting that there is not much difference in the vibrational kinetic energy 


1240 


with and without control state. It means that the acoustic actuator is able to 
reduce sound field globally without increasing plate vibration. 


3.2.3 control using both the piston source and the structural actuator 

In this section, a hybrid approach, simultaneous use of both the point force 
actuator and the acoustic piston source, is applied. The optimal strength of the 
force actuator and the piston source can be obtained from Eq. (26). Figure 5 
shows the acoustic potential energy of the cavity and the vibrational kinetic 
energy of the plate with and without the control actuators. Even at the cavity- 
controlled modes, it can be seen that a large reduction in the acoustic potential 
energy is achieved without significantly increasing the structural vibration. In 
general, more control actuators result in better control effects[6]. However, the 
hybrid approach with both structural and acoustic actuators in the system does 
not merely mean an increase in the number of actuators. As demonstrated in 
the last two sections, a single structural actuator is effective in controlling 
well separated plate-controlled modes and a single acoustic actuator is 
effective for controlling well separated cavity-controlled modes. Since the 
acoustic response is governed by both plate-controlled and cavity-controlled 
modes, the hybrid control approach can be desirable for controlling sound 
transmission in a ‘weakly-coupled’ structural acoustic system. 

To investigate the control efforts of each control system, the amplitude of the 
force actuator and the source strength of the piston source are plotted in 
Figure 6. There is a large decrease of the force amplitude at the well 
separated cavity-controlled modes, e.g. 85 Hz and 170 Hz, by using the both 
actuators. This trend can also be seen in the case of the piston source strength, 
especially at the 1st and 2nd structural natural frequency (52 Hz, 64 Hz). By 
using the hybrid approach, simultaneous use of both actuators, better control 
effects on the vibration of the plate, the transmission noise reduction and the 
control efforts of the actuators can be achieved. 


4. Conclusion 

The active control of the sound transmission into a ‘weakly coupled’ 
structural-acoustic system has been considered. The results obtained 
demonstrates that a single point force actuator is effective in controlling well 
separated plate-controlled modes, whereas, a single acoustic piston source is 
effective in controlling well separated cavity-controlled modes. 

By using the hybrid approach with both structural and acoustic actuators, 
improved control effects on the plate vibration, further reduction in sound 


1241 


transmission, and reduced control efforts of the actuators can be achieved. 

Since the acoustic behaviour is governed by both plate and cavity resonances, 

the hybrid control approach can be desirable in controlling sound transmission 

in a ‘weakly coupled’ structural-acoustic system. 

References 

1. C.R. FULLER, S.J. ELLIOTT and P.A. NELSONActive control of 
vibration, Academic Press Limited, 1996 

2. P.A. NELSON and S.J. ELLIOTT Active control of sound, Academic 
Press Limited, 1992 

3. B.-T. WANG, C.R. FULLER and K. DIMITRIADIS Active control of 
noise transmission through rectangular plates using multiple 
piezoelectric or point force actuators Journal of the Acoustical Society 
of America, 1991, 90(5), 2820-2830. 

4. M.E. JOHNSON and S. J. ELLIOTT Active control of sound radiation 
using volume velocity cancellation. Journal of the Acoustical Society of 
America, 1995, 98(4), 2174-2186. 

5. C.R. FULLER, C.H. HANSEN and S.D. SNYDER Active control of 
sound radiation from a vibrating rectangular panel by sound sources 
and vibration inputs: an experimental comparison, Journal of Sound 
and Vibration, 1991,145(2), 195-215. 

6. L. MEIROVrrCH and S. THANGJITHAM Active control of sound 
radiation pressure, Trans, of the ASMS Journal of Vibration and 
Acoustics, 1990,112, 237-244. 

7. S.D. SNYDER and N. TANAKA On feedforward active control of 
sound and vibration using vibration error signals, Journal of the 
Acoustical Society of America, 1993, 94(4), 2181-2193. 

8. J. PAN C.H. HANSEN and D. A. BIES Active control of noise 
transmission through a panel into a cavity : I. analytical study. Journal 
of the Acoustical Society of America, 1990, 87(5), 2098-2108. 

9. P.M. MORSE and K.U. INGARD Theoretical Acoustics, McGraw- 
Hill, 1968 

10. E.H. DOWELL, G.F. GORMAN HI, and D.A. SMITH 
Acoustoelasticity : general theory, acoustic modes and forced response 
to sinusoidal excitation, including comparisons with experiment. 
Journal of Sound and Vibration, 1977, 52(4), 519-542. 

11. F. FAHY Sound and Structural Vibration, Radiation, Transmission 
and Response, Academic Press Limited, 1985 


1242 


incident plane wave 



Figure 1 A structural acoustic coupled system with the volume V and its flexible boundary 
surface S/. 



Figure 2 The rectangular enclosure with one simply supported plate on the surface Sf on 
which external plane wave is incident with the angles of (cp = 0°) and (0=45°). 


Table 1 Material properties 


Material 

Density 

Phase speed 

Young’s 

Poisson’s 

Damping 


(kg/m^) 

(m/s) 

modulus (N/m^) 

ratio (v) 

ratio (0 

Air 

1.21 

340 

- 

- 

0.01 

Steel 

7870 

- 

207x10® 

0.292 

0.01 


1243 









Table 2 The natural frequencies and geometric mode shape coupling coefficients of each 
uncoupled system 


Order: 


Plate, 

1 

2 

3 

4 

■„ , 5. 

7 

10 

■c 

’Type 


(1,1) 

(2,1):: 

0,1) : 

(4,1) 

(5,1) 

(6,1) 

(7,1) 

Cavity^ 


' Freq.- 

52 Hz 

64 Hz 

86Hz 

115 Hz 

154 Hz 

200 Hz 

;256Hz 

1 

(0,0,0) 

0 Hz 

0.71 

0 

0.24 

0 

0.14 

0 

O.IO 

2 

(1,0.0) 

85 Hz 

0 

0.67 

0 

0.27 

0 

0.17 

0 

3 

(2,0.0) 

170 Hz 

-0.33 

0 

0.60 

0 

0.24" 

0 

0.16 

4 

(0,0.1) 

189 Hz 

-1.00 

0 

-0.33 

0 

-0.20" 

0 

-0.14 

5 

(1,0,1) 

207.HZ 

0 

-0.94 

0 

-0.38 

0 

-0.24 

0 

6 

(2;0,1) 

254 Hz 

0.47 

0 

-0.85 

0 

-0.34 

0 

-0.22 

7 

(3,0,0) 

255 Hz 

0 

-0.40 

0 

0.57 

0 

0.22 

0 



1 oo ISO 200 2SO 

Frequency 

(a) the acoustic potential energy of the cavity 



TSO 

Frequency 


(b) the vibrational kinetic energy of the plate(dB ref.= 10 ^ J) 

Figure 3 Effects of minimising the acoustic potential energy using a point force actuator ( 
solid line : without control, dashed line : with control ), where **’ and ‘o’ are at uncoupled 
plate and cavity natural frequencies, respectively. 




301 




Frequency 

(a) the acoustic potential energy of the cavity 



Figure 4 Effects of minimising the acoustic potential energy using an acoustic piston source 
( solid line ; without control, dashed line : with control), where and ‘o’ are at uncoupled 
plate and cavity natural frequencies, respectively. 



Figure 5 Effects of minimising the acoustic potential energy using both a point force 
actuator and an acoustic piston source - continued 


2 






Frequency 

(b) the vibrational kinetic energy of the pIate(dB ref=10‘^ J) 


Figure 5 Effects of minimising the acoustic potential energy using both a point force 
actuator and an acoustic piston source ( solid line : without control, dashed line: with control 
), where and ‘o’ are at uncoupled plate and cavity natural frequencies, respectively. 



Frequency 

(a) the strength of the force actuator 





Frequency 

(b) the strength of the piston source( unit: mVsec) 

Figure 6 Comparison of control efforts of the three control strategies; using each actuator 
separately ( solid line ) and using both the force actuator and the piston source (dashed line ) 
, where and ‘o’ are at uncoupled plate and cavity natural frequencies, respectively. 


1246 






A DISTRIBUTED ACTUATOR FOR THE 
ACTIVE CONTROL OF SOUND 
TRANSMISSION THROUGH A PARTITION 


TJ. Sutton, M.E. Johnson and S.J. Elliott 

Institute of Sound and Vibration Research 
University of Southampton, Southampton S017IBJ 


ABSTRACT 

The paper considers the problem of active control of soimd transmission 
through a partition using a single distributed actuator. The use of shaped, 
distributed actuators rather than point sources or locally-acting piezoceramic 
elements offers the possibility of controlling the volume velocity of a plate 
without giving rise to control spillover and avoids an increase in the sound 
radiated by uncontrolled structural modes. Specifically, a form of distributed 
piezoelectric actuator is described in which the electrode takes the form of a 
set of quadratic strips and serves to apply a roughly uniform normal force 
over its surface. 


INTRODUCTION 

The strong piezoelectric properties of the polymer polyvinylidene fluoride 
(PVDF) were discovered in 1969 [1]. The material is lightweight, flexible, 
inexpensive and can be integrated into engineering structures for strain 
sensing and to apply distributed forces and moments for the active control of 
vibration and sound transmission. Such ^smarf materials offer the possibility 
of providing lightweight sound-insulating barriers for application to aircraft, 
ground-based transport and in buildings. 

Lee [2] has set out the underlying theory of active laminated structures in 
which one or more layers of flexible piezoelectric material are attached to a 
plate. Practical sensors using PVDF material have been implemented by 
Clark and Fuller [3], Johnson and Elliott [4-6], and others. In these cases thin 
PVDF films were attached to the structure to sense integrated strain over a 
defined area. In [4] for example a distributed sensor was developed whose 
output is proportional to the integrated volume velocity over the surface of a 
plate. 


1247 


A number of studies have been carried out in which distributed piezoelectric 
actuators form a layer of a laminated system. In [7] the shape of a distributed 
piezoelectric actuator was chosen to be orthogonal to all but one of the natural 
modeshapes of the cylindrical shell system xmder control. Using this 
approach a set of actuators could be matched to the modes of the system 
under control, avoiding control spillover {i.e. the excitation of tmcontrolled 
structural modes). 

In the present paper a single shaped PVDF actuator is applied to a thin plate 
to control the noise transmission through it. The shape of the actuator is 
chosen specifically to apply an approximately uniform force to the plate. 
Such an actuator can be used to cancel the total volume velocity of the plate 
and therefore substantially to reduce the radiated sound power. (If volume 
velocity is measured at the plate surface there is no requirement for a remote 
error microphone.) As noted by Johnson and Elliott [6], the soimd power 
radiated by a plate which is small compared with an acoustic wavelength 
depends mainly on the volume velocity of the plate. The simulations in [6] 
show that provided the plate is no larger than half an acoustic wavelength, a 
single actuator used to cancel volume velocity will achieve similar results to a 
strategy in which radiated power is minimised. It is possible to envisage a 
large partition made up of a number of active plate elements designed on this 
basis. 


CALCULATION OF NORMAL FORCES IN THE PLATE 

In this section the equation of motion of the plate and attached PVDF layers is 
set out. The film thickness is assumed to be 0.5 mm. The analysis broadly 
follows that of Dimitriadis, Fuller and Rogers [8], but the individual 
piezoelectric coefficients and included separately as is appropriate 

for PVDF and a sensitivity function is included to account for variations in 
electrode shape. In addition the bending stiffness of the piezoelectric film is 
included (it is not negligible as the whole plate is covered). The nomenclature 
matches that used by Fuller, Elliott and Nelson [9] but here the analysis leads 
to the inhomogeneous wave equation for the plate-actuator system. 

We consider an aluminium plate of thickness Ih^ as shown in Figure 1. The 
plate is covered on its upper and lower surfaces with a piezoelectric film of 
sensitivity (}>(x,y)d-p in which d.^ is the strain/electric field matrix of the material 
(3x6 array) and <^{x,y) is a spatially-varying sensitivity function 
(0 < (j)(x,y) < 1). The two piezoelectric films are assumed to be identical but 
the same drive voltage is applied with opposite polarity to the lower film. As 
a result of this antisymmetric arrangement, the plate is subject to pure 
bending with no straining of the plate midplane. 


1248 


In line with other similar calculations [2,9] it is assumed that any line 
perpendicular to the midplane before deformation will remain perpendicular 
to it when the plate/PVDF assembly is deformed. As a result, the strain at 
any point in the assembly is proportional to distance z through it. (z=0 is 
defined to be on the midplane of the plate as indicated in Figure 1.) The 
direct and shear strains throughout the whole assembly (e^, Ey, E^y) are then 
given by [10]: 


3^^ 




d^w 




(1-3) 


in which w is the displacement of the midplane in the z direction. 

The corresponding stresses in the plate (only) follow from Hooke's law as in 
the standard development for thin plates: 


p __ p_ 


= 


i-v: 




A ^ p 

0'' =—— - 


2a+vj 


(4-6) 


in which is the Young's modulus and Vp is Poisson's ratio for the plate 
material. Stress in the piezoelectric film follows from the constitutive 
equations for the material [2]. The direct and shear stresses for the upper 
piezo film are designated of of and respectively: 




's/ 



K 

or"’ 

= C ^ 






1 - 

CO 


_^36. 


in which is the voltage applied across the actuator fibn (thickness /rj and 
4 and are the strain/field coefficients for the material. For PVDF d^, = 
0, but it is included in the analysis for completeness. As in [2], the stiffness 
matrix C is given by: 


1249 



0 


c 


^pe ^ pe^pe 

1 — 

^ ^ pe ^ pe 

^ pe^pe ^pe 

1-v' 1-v^ 

0 0 


0 

^pe 

2(l + v^J 


( 8 ) 


The stresses in the lower piezoelectric layer are designated and 

, and the form of the expression is similar to the upper layer except that 
the voltage is applied with reversed polarity: 




's/ 


^31 

V 


= c 


+ (^{x,y) 








a 


Summing moments about the x and y axes for a small element dxby of the 
plate yields the moment per unit length about the y-axis and My about 
the x-axis; also the corresponding twisting moments per imit length, My^ and 


M^=jc!’’zdz+ 

-III, ~K-K ^‘b 

(10) 

h,, -III, ‘‘b+lh, 

My = j a^yZdz + J + J cf'^zdz 

-III, -hh-K h 

(11) 

h,, -III, l>h+lh, 

= |<t;,z*+ ja^fhdz 

(12) 


-III, -K-K h 


and My^=M,y. 


The vertical acceleration at each point of the plate d^w/dt^ is obtained by 
taking moments about the x and y axes for a small element 6x6y and 
resolving vertical forces as in standard thin plate theory. If the plate is acted 
on by some external force per unit area p(x,y)f(t) then the vertical motion of 

the plate is described by: 


dx^ dxdy dy^ dt^ 


= -p(x,y)f(t) 


(13) 


1250 


in which m is the mass/area of the plate-film assembly. 


The equation of motion of the plate complete with attached piezoelectric film 
is obtained by combining the above equations. For convenience the following 
constants are defined; 


n _ 

'' 3(1 


(bending stiffness of plate, thickness 2/i,) (14) 


3(1-V^) 


(bending stiffness due to PVDF) (15) 


+ 2 / 1 ,) 


The equation of motion of the plate assembly including upper and lower 
piezoelectric layers is then given by: 




3^(l)(^,y) 


3>(j:,y) 


The left-hand side of this equation determines the free response of the plate- 
film assembly and is recognised as the standard form for a thin plate. The 
first term on the right-hand side is the assumed externally applied normal 
force per imit area. The second term on the right-hand side gives the effective 
normal force per unit area applied to the plate due to the two piezoelectric 
films driven by a voltage . It is clear that this force depends on the 
spatially-varying sensitivity which has been assumed for the piezoelectric 
material. 

An examination of Eqns. (16) and (17) shows that the normal force applied by 
the piezoelectric film depends on the sum of the plate thickness and the 
thickness of one of the film layers. If the film is much thinner than the plate 
(h^« hy) then the applied force becomes independent of the film thickness and 
depends only on the plate thickness, the applied voltage, the electrode pattern 
and the material constants. Eqn. (17) also shows that the normal force is 
applied locally at all points on the plate. No integration is involved, and so in 
contrast with a volume velocity sensor designed using quadratic strips [5], the 
force does not depend on the plate boundary conditions in any way. 
Furthermore, there is no need to use two films oriented at 90 degrees to cancel 

1251 


the cross-sensitivity It is also worth noting that no assumption has been 
made about the modeshapes on the plate. 

We can create a uniform force actuator by choosing: 

^ = constant, and (18) 

Bx 

(19) 

This can approximately be achieved by depositing electrodes in the form of 
narrow strips whose width varies quadratically in the x-direction. (Note that 
the x-direction is defined as the direction of rolling of the PVDF material, Le. 
the direction of for maximum sensitivity.) The form of the electrodes is as 
shown in Figure 2. With this pattern the sensitivity function takes the form: 

(t)(x,>') = (20) 

where is the length of the strip. Thus (t)(-^>^) - ^ at x = 0 and x = (no 
electrode), while (|)(x, 31 ) = 1 halfway along at x = LJ2 (electrode fully covers 
the film). 


APPLICATION TO A THIN ALUMINIUM PLATE 

If a plane wave of sound pressure level 94 dB (say) is normally incident on a 
hard surface, it will exert a pressure of 2 Pa rms on that surface. If this 
incident pressure is counterbalanced by a uniform force actuator applied to a 
plate, then the plate could in principle be brought to rest. Thus for active 
control of everyday noise levels the uniform force actuator will need to be 
able to generate a normal force/area of a few pascal over the surface of the 
plate. (When the incident wave impinges on the plate at an oblique angle, 
many natural modes of the plate will be excited and it will not be possible to 
bring it perfectly to rest with a single actuator; however it will remain 
possible to cancel the plate volume velocity as explained earlier.) 

By way of example an aluminium plate of thickness 1 mm will be assumed, 
with a free surface measuring 300 x 400 mm. Attached to each side is a PVDF 
film of thickness 0.5 mm. One electrode of each panel would be masked to 
give quadratic strips of length 300 mm as shown in Figure 2. (The width of the 
strips is unimportant, but should be significantly smaller than the structural 
wavelength of modes of interest on the plate.) In this case it turns out that 

= 6.64 for the plate, and 


1252 


Dp, = 1.27 for the PVDF film. 

The piezoelectric constants for the film are typically 

4 = 23 X m/V and 

4 = 3 X lO’"' m/V, giving 

Cp, = 3.28x10^ 

Finally the double derivative of the sensitivity function turns out to be 

= 88.9 (300 mm strip length) 

dx^ 

The bending stiffness of the 1 mm aluminium plate is increased by 20% due to 
the addition of two layers of PVDF film of thickness 0.5 mm each. The force 
per unit area due to the actuator is obtained from Eqn. (17): 

force/area = 4 ,(^ 3 ,+Vp, 4)|^'^3 

= 6.96x10’^ V 3 Pa 

Thus 1000 volt rms would yield a tmiform force/area of close to 7 Pa. This is 
not an impracticable voltage level, but previous experience at ISVR suggests 
that care would need to be taken to avoid electrical breakdown through the 
air between electrodes, or over damp surfaces. 


ACTIVE CONTROL OF SOUND TRANSMISSION 

In reference [ 6 ] Johnson and Elliott have presented simulations of the active 
control of harmonic sound transmitted through a plate using a uniform force 
actuator. Their actuator might be realised along the lines described in this 
paper. In the simulatioi\s presented in [ 6 ] the uniform force actuator is used 
with a matched volume velocity sensor having the same electrode shape [4]. 
The advantage of this configuration is that the actuator can be used to drive 
the net volume velocity of the plate to zero without exciting high order 
structural modes in the process (control spillover). Simulations of a 
300x380x1 mm aluminium plate showed that reductions in transmitted sound 
power of around 10 dB were achievable in principle up to 600 Hz using this 
matched actuator-sensor arrangement. 

A further advantage of the distributed matched actuator-sensor pair is that 
the secondary path through the plate (for active control) is minimum phase 
[ 6 ], giving good stability characteristics if a feedback control loop is 
implemented to control random incident soimd for which no reference signal 
is available. 


1253 


CONCLUSION 


A design of distributed piezoelectric actuator has been presented which 
generates a roughly uniform force over the surface of a plate. An example 
calculation shows that the design is capable of controlling realistic soimd 
pressure levels. When used in combination with a matched volume velocity 
sensor, the actuator-sensor pair will have minimum-phase characteristics and 
will offer the possiblity of feedback control in which neither a reference signal 
nor a remote error sensor will be required. 


ACKNOWLEDGEMENT 

The financial support of the European Community under the Framework IV 
programme is gratefully acknowledged. (Project reference: BRPR-CT96-0154) 


REFERENCES 

[1] G.M. Sessler (1981) JAcoust Soc Am 70(6) Dec 19811596-1608 
Piezoelectricity in polyvinylidene fluoride 

[2] C.K. Lee (1990) JAcoust Soc Am 87(3) Mar 1990 1144-1158 Theory of 
laminated piezoelectric plates for the design of distributed sensors/actuators. 
Part I: Governing equations and reciprocal relationships 

[3] R.L. Clark and C.R. Fuller (1992) JAcoust Soc Am 91(6) June 1992 3321-3329 
Modal sensing of efficient acoustic radiators with polyvinylidene fluoride 
distributed sensors in active structural acoustic control approaches 

[4] M.E. Johnson, S.J. Elliott and J.A. Rex (1993) ISVK Technical Memorandum 
723. Volume Velocity Sensors for Active Control of Acoustic Radiation 

[5] M.E. Johnson and S.J. Elliott (1995) Proceedings of the Conference on Smart 
Structures and Materials 27 Feb-3 Mar 1995, San Diego, Calif. SPIE Vol 2443. 
Experiments on the active control of sound radiation using a volume velocity 
sensor 

[6] M.E. Johnson and S.J. Elliott (1995) JAcoust Soc Am 98(4) Oct 1995 2174- 
2186. Active control of sound radiation using volume velocity cancellation 

[7] H.S. Tzou, J.P. Zhong and J.J. Hollkamp (1994) Journal of Sound and 
Vibration 177(3) 363-378 Spatially distributed orthogonal piezoelectric shell 
actuators: theory and applications 

[8] E.K. Dimitriadis, C.R. Fuller and C.A. Rogers (1991) Transactions of the 
ASME, Journal of Vibration and Acoustics 113 100-107 Piezoelectric actuators for 
distributed vibration excitation of thin plates 

[9] C.R. Fuller, S.J. Elliott and P.A. Nelson (1996) Active Control of Vibration. 
Academic Press, London. 

[10] G.B. Warburton (1976) The Dynamical Behaviour of Structures, 2nd Edition. 
Pergamon Press, Oxford. 


1254 




layers of 
PVDF film 


Figure 1: Schematic diagram of thin plate 
covered on both sides with a layer of PVDF film 



Figure 2: Electrode pattern of quadratic strips for 
uniform-force actuator 


1255 











1256 


CONTROL OF SOUND RADIATION FROM A FLUID-LOADED PLATE 
USING ACTIVE CONSTRAINING LAYER DAMPING 

J. Ro, A. Al-Ali and A. Baz 

Mechanical Engineering Department 
The Catholic University of America 
Washington D. C. 20064 

Abstract 

Sound radiation from a vibrating flat plate, with one side subjected to 
fluid-loading, is controlled using patches of Active Constrained Layer Damping 
(ACLD). The fluid-structure-controller interaction is modeled using the finite 
element method. The damping characteristics of the ACLD/plate/fluid system are 
determined and compared with the damping characteristics of plate/fluid system 
controlled with conventional Active Control (AC) and/or Passive Constrained 
Layer Damping (PCLD) treatments. Such comparisons are essential in 
quantifying the individual contribution of the active and passive damping 
components to the overall damping characteristics, when each operates separately 
and when both are combined to interact in unison as in the ACLD treatments. 

I. INTRODUCTION 

When a structure is in contact with or immersed in a fluid, its vibration 
energy radiates into the fluid domain. As a result, there is an observable increase 
in the kinetic energy of the structure due to the fluid loading. Because of this 
kinetic energy increase, the natural frequencies of structures which are subjected 
to fluid-loading decrease significantly compared to the natural frequencies of 
structures in vacuo. Therefore, through understanding of the interaction between 
the elastic plate structures and the fluid loading has been essential to the effective 
design of complex structures like ships and submarine hulls. Lindholm et al. [1] 
used a chordwise hydrodynamic strip theory approach to study the added mass 
factor for cantilever rectangular plates vibrating in still water. Fu et al [2] studied 
the dry and wet dynamic characteristics of vertical and horizontal cantilever 
square plates immersed in fluid using linear hydroelasticity theory. Ettouney et 
al [3] studied the dynamics of submerged structures using expansion vectors, 
called wet modes which are finite series of complex eigenvectors of the fluid- 
structure system. Recently Kwak [4] presented an approximate formula to 
estimate the natural frequencies in water from the natural frequencies in vacuo. 

When the structure and the fluid domains become rather complex, 
solutions of fluid-structure coupled system can be obtained by finite element 


1257 


methods. Marcus [5], Chowdhury [6], Muthuveerappan et al. [7] and Rao et al 
[8] have successfully implemented the finite element method to predict the 
dynamic characteristics of elastic plates in water. Everstine [9] used both finite 
and boundary element methods to calculate the added mass matrices of fiilly- 
coupled fluid-structure systems. 

The above investigations formed the bases necessary to devising passive 
and active means for controlling the vibration of as well as the sound radiation 
from fluid-loaded plates. Passive Constrained Layer Damping (PCLD) treatments 
have been used extensively and have proven to be effective in suppressing 
structural vibration as reported, for example, by Jones and Salerno [10], Sandman 
[11] and Dubbelday [12]. Recently, Gu and Fuller [13] used feed-forward control 
algorithm which relied in its operation on point forces to actively control the 
sound radiation from a simply-supported rectangular fluid-loaded plate. 

In the present study, the new class of Active Constrained Layer Damping 
(ACLD) treatment is utilized as a viable alternative to the conventional PCLD 
treatment and Active Constrained (AC) with PCLD treatment (AC/PCLD). The 
ACLD treatment proposed combines the attractive attributes of both active and 
passive damping in order to provide high energy dissipation-to-weight 
characteristics as compared to the PCLD treatment. Such surface treatment has 
been successfully employed to control the vibration of various structural members 
as reported, for example, by Shen [14] and Baz and Ro [15]. In this paper, the use 
of the ACLD is extended to the control of sound radiation from fluid-loaded 
plates. Finite element modeling of the dynamics and sound radiation of fluid- 
loaded plates is developed and validated experimentally. Particular focus is 
placed on demonstrating the effectiveness of the ACLD treatment in suppressing 
the structural vibration and attenuating the sound radiation as compared to 
conventional PCLD and AC/PCLD. 

This paper is organized in five sections. In Section 1, a brief introduction 
is given. In Section 2., the concepts of the PCLD, ACLD and AC/PCLD 
treatments are presented. In Section 3, the dynamical and fluid finite element 
models are developed to describe the interaction between the plate, ACLD and the 
contacting fluid. Experimental validation of the models are given in Section 4. 
Comparisons between the theoretical and experimental performance are also 
presented in Section 4 for different active and passive damping treatments. 
Section 5, summarizes the conclusions of the present study. 

2. CONCEPTS OF PCLD, ACLD AND AC/PCLD TREATMENTS 

Figures (1-a), (1-b) and (1-c) show schematic drawings of the PCLD, 
ACLD and AC/PCLD treatments respectively. In Figure (1-a), the plate is treated 


1258 






Figure (1) - Schematic drawing of different surface treatments (a) PCLD, (b) 
ACLD and (c) AC/PCLD. 


with a viscoelastic layer which is bonded directly to the plate. The outer surface 
of the viscoelastic layer is constrained by an inactive piezo-electric layer in order 
to generate shear strain y, which results in dissipation of the vibrational energy of 
the plate. Activating the constraining layer electrically, generates a control force 
Fp by virtue of the piezo-electric effect as shown in Figure (l-b) for the ACLD 
treatment . Such control action increases the shear strain to yj which in turn 
enhances the energy dissipation characteristics of the treatment. Also, a restoring 
moment Mp=d 2 Fp is developed which attempts bring the plate back to its 
undeformed position. In the case of AC/PCLD treatment, shown in Figure (l-c), 
two piezo-films are used. One film is active and is bonded directly to the plate to 
control its vibration by generating active control (AC) force Fp and moment 
Mp^djFp. The other film is inactive and used to restrain the motion of the 


1259 




viscoelastic layer in a manner similar to the PCLD treatment of Figure (1-a). In 
this way, the AC action operates separately from the PCLD action. This is unlike 
the ACLD configuration where the active and passive control actions operate in 
unison. Note that in the ACLD configuration, larger shear strains are obtained 
hence larger energy dissipation is achieved. Furthermore, larger restoring 
moments are generated in the ACLD treatments as compared to the AC/PCLD 
treatments as the moment arm d 2 in the former case is larger than the moment arm 
d 3 of the latter case. This results in effective damping of the structural vibrations 
and consequently effective attenuation of sound radiation can be obtained. 


3. FINITE ELEMENT MODELING 

3.1 Overview 

A finite element model is presented in this section, to describe the 
behavior of fluid-loaded thin plates which are treated with ACLD, PCLD and 
AC/PCLD treatments. 

3.2 Finite Element Model of Treated Plates 




(b) (c) 

Figure (2) - Schematic drawing of plate with ACLD/AC/PCLD patches. 










Figure (2) shows a schematic drawing of the ACLD and AC/PCLD 
treatments of the sandwiched plate which is divided into N finite elements. It is 
assumed that the shear strains in the piezo-electric layers and in the base plate are 
negligible. The transverse displacement w of all points on any cross section of the 
sandwiched plate are considered to be the same. The damping layers are assumed 
to be linearly viscoelastic with their constitutive equations described by the 
complex shear modulus approach such that G=G ( 1 +T|i). In addition, the bottom 
piezo-electric layer (AC) and the base plate are considered to be perfectly bonded 
together and so are the viscoelastic layer and the top piezo-electric layer. 


The treated plate elements considered are two-dimensional elements 
bounded by four nodal points. Each node has seven degrees of fireedom to 
describe the longitudinal displacements u, and v, of the constraining layer, U 3 and 
V 3 of the base plate, the transverse displacement w and the slopes w and W y of 
the deflection line. The deflection vector {5} can be written as: 

{5} = {u„v„U3, V3,W, W ^W y}"" 

= [{n,} {N 3 } {Nj {n.} {n,} {N,}^ 

where {5"} is the nodal deflection vector, {Nj}, {Nj}, {N 3 }, {N 4 }, {N 5 }, {N 5 },,, 
and {N 5 } y are the spatial interpolating vectors corresponding to u„ v„ U 3 , V 3 , w, 
w^, and Wy respectively. Subscripts ,x and ,y denote spatial derivatives with 
respect to x and y. 

Consider the following energy functional ITp for the treated plate/fluid 

system: 

np = I(u-TK+w.-w^+wJdv, ( 2 ) 

where U is the strain energy, T^ is the kinetic energy, is work done by external 
forces, Wp is work done by the back pressure inside the fluid domain, is work 
done by the control forces and moments and V is the volume of the plate. These 
energies are expressed as follows 



I T,dv =X ^ 1 1„ (*' = 5{®T['^p]{®') ’ 

|w,dV=j{5'f{F}, 


= 5' Kj 5' 


“ V.l^dxdy 

„[Ui hi ‘J “ ax^ Ui ‘ hi j “ Sy^J 

where i=l for ACLD control or i=3 for AC control ( 6 ) 


and I WpdV ={5'}’"[n ]{p'}. 


where {p®} is the nodal pressure vector of the fluid element. In the above 
equation []^], [Mp], {F}, [KJ and [Q] are the plate stiffness matrix, mass matrix, 
external forces vector, piezo-electric forces and moments matrix and plate/fluid 
coupling matrix as given in the appendix. In equation ( 6 ), d 3 i 32 are the piezo¬ 
strain constants in directions 1 and 2 due to voltage applied in direction 3. The 
voltage is generated by feeding back the derivative of the displacement 5 at 

critical nodes such that j where is the derivative feedback gain 

matrix and C is the measurement matrix defining the location of sensors. 

Minimizing the plate energy fimctional using classical variational methods 

such that |anp/a{6®}j = 0 leads to the following finite element equation: 

{[K]-»lM.]){5'}-[n]{p'} = {F} ( 8 ) 

where co is the frequency and [K] = [Kp] + [K^.] is overall stiffriess matrix. 

3.3 Finite Element Model of the Fluid 

The fluid model uses solid rectangular tri-linear elements to calculate the 
sound pressure distribution inside the fluid domain and the associated structural 
coupling effects. The fluid domain is divided into fluid elements. Each of 


1262 


these elements has eight nodes with one degree of jfreedom per node. The 
pressure vector is expressed by p = [Nf]{p®} and [NJ is pressure shape function 
and {p®} is nodal pressure vector. 

Considering the following functional Elf of fluid domain Craggs [16] 

where [Kf] and [MJ are the fluid stiffiiess and mass matrices as given in the 
appendix. Minimization of equation (9) such that {OTf/^lp^}} = 0 yields the 
fluid dynamics as coupled with the structural vibration: 

([K,]-o)^[M,]){p'}-<a^[nf{5'} = {0} (10) 

The boundary conditions involved are of the form 

ap/an = 0, at a rigid boundary 


a p / a n = -pp 9^5 / 9, 
and p = 0. 


at a vibrating boundary 
at a free surface 


where pf is fluid density. 

3.4. Solutions of the Coupled Plate/Fluid System 

Combining equations (8) and (10) gives 

■[K]-co^[m,] -[fi] Ip'l.pl (11) 

[Kr]-CD^[M,]J Lp'J k 

At low frequencies, the fluid pressure is in phase with the structural 
acceleration, i.e. the fluid appears to the structure like an added mass. However, 
as the frequency increases the added mass effect diminishes and the damping 
effect, i.e. the pressure proportional to velocity, increases. For an incompressible 
fluid, the speed of sound c approaches oo, thus the mass matrix of the fluid [MJ 
vanishes, and equation (11) can be simplified to 

■[K]-cd=[m,] -[fl]] rs'] pi (12) 

_ [KfiJkJ k 

If the fluid-structure coupled system has free boundary surface, then [KJ is 
non-singular [Everstin, 1991] and the nodal pressure vector {p®} can be eliminated 
from equation (12) as follows: 

{p'}=-o.^[K,nnr{5'} (13) 


1263 


Hence, equation (12) yields 


([K]-ffl^(K]+[Mj)){8') = {F) (14) 

where [MJ is added virtual mass matrix defined by [Muthuveerappan, 1979] 

[Mj = [a][K,]-'[£2f (15) 

Equation (14) only involves the unknown nodal deflection vector {5®} of the 
structure. When {F}=0, equation (14) becomes an eigenvalue problem, the 
solution of which yields the eigenvalues and eigenvectors. The nodal pressures 
can then be obtained from equation (13) when the nodal displacements are 
determined for any particular loading on plate. 

4. PERFORMANCE OF PARTIALLY TREATED PLATES WITH 
FLUID LOADING 

In this section, comparisons are presented between the numerical 
predictions and experimental results of the natural frequencies and damping ratios 
of a fluid-loaded plate treated with ACLD, PCLD and AC/PCLD. The effect of 
the Active Control, Passive Constrained Layer Damping and Active Constrained 
Layer Damping on the resonant frequency, damping ratio, attenuation of vibration 
amplitude and sound radiation are investigated experimentally. The vibration and 
sound radiation attenuation characteristics of the fluid-structure coupled system 
are determined when the plate is excited acoustically with broadband frequency 
excitation while the piezo-electric layers are controlled with various control gains. 
The experimental results are compared with the theoretical predictions. 

4.1 Experimental Set-up 

Figures (3-a) and (3-b) show a schematic drawing and finite element mesh 
of the experimental set-up along with the boundary conditions used to describe 
the fluid-structure system. The finite element mesh includes: 24 plate-elements 
and 560 fluid-elements. The coupled system has a total of 815 active degree of 
freedoms. The aluminum base plate is 0.3m long, 0.2m wide and 0.4inm thick 
mounted with all its edges in a clamped arrangement in a large aluminum base. 
The aluminum base with mounting frame sits on top of a water tank. One side of 
the base plate is partially treated with the ACLD/AC/PCLD and the other side is 
in contact with water. The material properties and thickness of piezo-electric 
material and the viscoelastic layer listed in Table (1). The size of the combined 
piezo-electric and viscoelastic patch occupied one-third of the surface area of the 
base plate and it is placed in the middle of plate as shown in Figure (3-b). A laser 


1264 



sensor is used to measure the vibration of the treated plate at node 27 as shown in 
Figure (3-b). The sensor signal is sent to a spectrum analyzer to determine the 
frequency content and the amplitude of vibration. The signal is also sent via 
analog power amplifiers to the piezo-electric layers to actively control the sound 
radiation and structural vibration. The radiated sound pressure level into the tank 
is monitored by a hydrophone located at 5.0 cm below the plate center. This 
position is chosen to measure the plate mode (1, 1) which dominates the sound 
radiation. The hydrophone signal is sent also to the spectrum analyzer to 
determine its frequency content and the associated sound pressure levels. 



Figure (3) - The experimental set-up, (a) schematic drawing, (b) finite element 
meshes. 


Table (1) - Physical and geometrical properties of the ACLD treatment 


Layer 

Thickness(m) 

Density (Kg/m^) 

Modulus(MPa) 

viscoelastic 

S.OSxlO’" 

1104 

30** 

piezoelectric 

28xl0-‘ 

1780 



* Young’s modulus ** Shear modulus 


4.2 Experimental Results 

Experimental validation of the dynamic finite element model of the 
ACLD/plate system in air has been presented by Baz and Ro [15] in detail. Close 
agreement was obtained between the theoretical predictions and the experimental 


1265 












measurements. The d)mamic finite element model is therefore valid to provide 
accurate predictions. 

For the uncontrolled treated plate/fluid system, considered in this study, 
the experimental results indicate that coupling the plate with the fluid loading 
results in decreasing the first mode of vibration fi*om 59.475Hz to 10.52Hz. The 
coupled finite element model predicts the first mode of vibration to decrease firom 
57.91Hz to 10.24Hz. The results obtained indicate close agreement between the 
theory and experiments. 

Figure (4-a) shows a plot of the normalized experimental vibration 
amplitudes for the fluid-loaded plate with the ACLD treatment using different 
derivative feedback control gains. According to Figure (4-a), the experimental 
results obtained by using the ACLD treatment indicate that amplitude attenuations 
of 11.36%, 48.25% and 75.69% are obtained, for control gains of 2500, 5000, and 
13500, respectively. The reported attenuations are normalized with respect to the 
amplitude of vibration of uncontrolled plate, i.e. the plate with PCLD treatment. 
Figures (4-b) display the vibration amplitudes of the plate/fluid system with 
AC/PCLD treatment at different derivative feedback control gains. The 
corresponding experimental attenuations of the vibration amplitude obtained are 
4.6%, 20.29%, 54.04% respectively. 




Figure (4) - Effect of control gain on normalized amplitude of vibration of the 
treated plate, (a) ACLD control and (b) AC/PCLD control. 


Figures (5-a) and (5-b) show the associated normalized experimental 
sound pressure levels (SPL) using ACLD and AC/PCLD controllers, respectively. 
The normalized experimental SPL attenuations obtained using the ACLD 
controller are 26.29%, 50.8% and 76.13% compared to 10.02%, 24.52% and 
53.49% with the AC/PCLD controller for the considered control gains. Table (2) 


1266 





lists the maximum control voltages for the ACLD and AC/PCLD controllers for 
the different control gains. 

It is clear that increasing the control gain has resulted in improving the 
attenuations of the plate vibration and the sound radiation into the fluid domain. 
It is evident that the ACLD treatment has produced significant vibration and 
sound pressure level attenuation as compared to the attenuations developed by the 
AC/PCLD or PCLD treatments. It is also worth emphasizing that the ACLD 
treatment requires less control energy than the conventional AC/PCLD treatments 
to control the sound radiation from the plate. 



Figure (5) - Effect of control gain on normalized sound pressure level radiated 
from the treated plate, (a) ACLD control and (b) AC/PCLD control. 


Table (2) - Maximum control voltage for the ACLD/AC/plate system 


K. 


2500 

5000 

13500 

ACLD 

0 

21.75 V 

31.20V 

39.60V 

AC 



50.40V 

76.38V 


Figure (6) shows the mode shapes of the first four modes of the treated 
plate with and without fluid-loading as obtained experimentally using 
STARMODAL package. Figure (7) shows the corresponding theoretical 
predictions of the first four mode shapes. Close agreement is found between 
experimental measurement and theoretical predictions. 

Figure (8) presents comparisons between the theoretical and experimental 
natural frequencies and the loss factor of a plate treated with the ACLD and 
AC/PCLD for different control gains. Close agreement between theory and 


1267 













experiment is evident. Note also that increasing the control gain has resulted in 
increasing the damping ratio for both ACLD and AC/PCLD treatments. The 
comparisons emphasize the effectiveness of the ACLD treatment in acquiring the 
large damping ratio to attenuate the structural vibration and sound radiation. 



Figure (6) - Experimental results of first four mode shapes of treated plate (a) 
without fluid loading and (b) with fluid loading. 



Figure (7) - Theoretical predictions of first four mode shapes of treated plate (a) 
without fluid loading and (b) with fluid loading. 


5. SUMMARY 

This paper has presented theoretical and experimental comparisons 
between the damping characteristics of plates treated with ACLD and 


1268 


conventional AC with PCLD treatments. The dynamic characteristics of the 
treated plates when subjected to fluid loading is determined for different 
derivative control gains. The fundamental issues governing the performance of 
this class of smart structures have been introduced and modeled using finite 
element method. The accuracy of the developed finite element model has been 
validated experimentally. The effectiveness of the ACLD treatment in attenuating 
structural vibration of the plates as well as the sound radiated from these plates 
into fluid domain has also been clearly demonstrated. The results obtained 
indicate that the ACLD treatments have produced significant attenuation of the 
structural vibration and sound radiation when compared to PCLD and to AC with 
PCLD. Such favorable characteristics are achieved with control voltages that are 
much lower than those used with conventional AC systems. The developed 
theoretical and experimental techniques present invaluable tools for designing and 
predicting the performance of the plates with different damping treatments and 
coupled with fluid loading that can be used in many engineering applications. 


♦ PCLD ■ ACLD, K:d=l3500 □ AC, K:d= 13500 

• ACLD,Kd=2500 O AC, Kd=2500 
A ACLD,Kd=5000 A AC, Kd=5000 



Theoretical Natural Frequency (Hz) Theoretical Damping Ratio 

Figure (8) - Comparison between theoretical predictions and experimental results, 
(a) natural frequency, (b) damping ratio. 

ACKNOWLEDGMENTS 

This work is funded by The U.S. Army Research Office (Grant number 
DAAH-04-93-G-0202). Special thanks are due to Dr. Gary Anderson, the 
technical monitor, for his invaluable technical inputs. 


1269 




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2. Fu, Y. and Price, W. G., Interactions between a partially or totally immersed 
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8. Rao, S. N. and Ganesan, N., Vibration of plates immersed in hot fluids. 
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sandwich plate radiating into a fluid medium. Trans, of ASME, Journal of 
Engineering for Industry, 1965, 379-384. 

11. Sandman B. E., Motion of a three-layered elastic-viscoelastic plate under 
fluid loading. J. of Acoustical Society of America, 1975, 57(5), 1097-1107. 

12. Dubbelday, P. S., Constrained-layer damping analysis for flexural waves in 
infinite fluid-loaded plates. Journal of Acoustical Society of America, 1991, 
(3), 1475-1487. 

13. Gu, Y. and Fuller, C. R., Active control of sound radiation from a fluid- 
loaded rectangular uniform plate. Journal of Acoustical Society of America, 
1993, 93(1), 337-345. 

14. Shen, I. Y., Bending vibration control of composite plate structures through 
intelligent constrained layer treatments. Proc. of Smart Structures and 
Materials Conference on Passive Damping ed. C. Johnson, 1994, Vol. 2193, 
115-122, Orlando, FL. 


1270 


15. Baz, A. and Ro, J., Vibration control of plates with active constrained layer 
damping. Journal of Smart Materials and Structures, 1996, 5, 272-280. 

16. Craggs, A., The transient response of a coupled plate-acoustic system using 
plate and acoustic finite elements. Journal of Sound and Vibration, 1971, 
15(4), 509-528. 

APPENDIX 

1. Stiffness Matrix of the Treated Plate Element 

The stiffoess matrix [Kp]; of the ith element of the plate/ACLD system is 
given by Baz and Ro [15]: 

(A-1) 

where [K,l and [KJi denote the in-plane, shear and bending stifj&iesses of 
the ith element. These stiffoess matrices are given by: 

j = layer 1,2,and3 (A-2) 

and [K.,1 = I jjB,]"[Dj,][B,]dxdy j = layer 1,2and3 (A-4) 

with G 2 denoting the shear modulus of the viscoelastic layer and the matrices [BJ, 


b1 = :^ 


\({N2}-{N4)/d + {N,} 

(n,1 +fNj 




’ [Bb] = 



2{N5},„_ 


{Naj 

[B.] = 

{n. 




[®^p] = 


(N.},. + {N3},.+h{Ns},„ 

{n.},, + + {N2}^ +{n.) „ +h{N4^, 


1271 



- 1 

0 

_i 


1 - 

0 

Ei 

1 n 

and fj, 1 EA 

1 n 

1-vJ 

Vj 1 u 


Vj 1 u 


I_ 


' « '“Vi 


0 

0 

i' 


_o 0 V. 


. j=l,2and3 (A-5) 


where h = (hi-h 3)/2 and d = (h 2 +hi/ 2 +D) with D denoting the distance from the 
mid-piane of the plate to the interface with the viscoelastic layer. Also, Ij 
represent the area moment of inertia of the jth layer. 

2. Mass Matrix of the Treated Plate Element 

The mass matrix [Mp]; of the ith element of the plate/ACLD system is 
given by: 

(A-6) 

where [Mjp]j and [M^Jj denote the mass matrices due to extension and bending of 
the ith element. These matrices are given by 

[Mi,].=p,h,££({N,}^{N,}+{N3}''{N,})dxdy + p,h,[ [({N3)'"{N3} + {NX{N4))dxdy 

+ ip2h2 £ { +{N,}^{N,})dxdy 

and [ ], = (p,h, + P 3 h 3 + P 3 h 3 ) £ £ [ N 3 f [ N 3 ]dxdy (A-7) 

where {NJ = {N,}+{N 3 }+h{Ns },3 and {N,} = {NJ+{N 4 }+h{N 5},3 

3. Control Forces and Moments Generated by the Piezo-electric Layer 

3.1 The in-plane piezo-electric forces 

The work done by the in-plane piezo-electric forces {Fp}i of the ith 
element is given by: 

i{5'}-{Fp}rhi££%d>'dy (A- 8 ) 


where j=l for ACLD control or j=3 for AC control. Also, Ojp and Sjp are the in¬ 
plane stresses and strains induced in the piezo-electric layers. Equation (A- 8 ) 
reduces to: 


1272 


='"•1IKFK] 


for k=l,4 


3.2 The piezo-electxic moments 

The work done by the piezo-electric moments {Mp}i due to the bending of 
the piezo-electric layer of the ith element is given by: 


5' .MM 


= hj U,e,Ady 


(A-10) 


Where Gjb and are the bending stresses and strains induced in the piezo-electric 
layers. Equation (A-10) reduces to: 


Mpxk 


x.' 

Mpyk 


d32 

Mp,yk 


_ 0 


fork=l,..,4 


(A-11) 


4. Stiffness and Mass Matrices of the Fluid Element 

The stiffness matrix [KJj and mass matrix [MJ^ of the ith element of the 
fluid system are given by: 


[B,rNdv 


(A-12) 

(A-13) 


[N,]>,]dv (A-13) 

where [b^] = [[N^],, [N^] ^ [N^] and c is the sound speed. 

5. Coupling matrix of the Treated Plate/Fluid System 

The coupling matrix [Q] of the interface element of the structure/fluid 
system can be presented by: 

[ q ]= f [ {N5}''[Nf]dxdy (A-14) 

Ja Jb 


1273 



1274 


ANALYTICAL METHODS II 



DYNAMIC RESPONSE OF SINGLE-LINK FLEXIBLE 
MANIPULATORS 


E. Manoach^ G. de Paz^ K. Kostadinov^ and F. Montoya^ 

^ Bulgarian Academy of Sciences, Institute of Mechanics 
Acad. G. Bonchev St. Bl. 4; 1113 Sofia, Bulgaria 
^ Universidad de Valladolid, E.T.S.I.L Dpto. IMEIM. 

C/Paseo del Cauce, s/n 47011- Valladolid, Spain 

1. INTRODUCTION 

The flexible-link manipulators have many advantages over the traditional stiff 
ones. The requirements for light-weight and energy efficient robotic arms 
could be naturally satisfied by using flexible manipulators. On the other hand 
the application of the robotic arm in such activities as positioning in electronic 
microscopes and disc-drivers, hammering a nail into a board or playing tennis, 
also forces the modeling and control of the dynamic behavior of flexible link 
manipulators. 

In most cases the elastic vibrations which arise during the motion must be 
avoided when positioning the end point of a robotic arm. These are a part of 
the reasons that cause a great increase of the publications in this topic in recent 
years. 

In most papers the flexible robotic arms are modeled as thin linear elastic 
beams. In [1-3] (and many others) the Bemouli-Euiler beam theory, combined 
with finite-element technique for discretization with respect to the space 
variables is used for modeling and control of single-link flexible manipulators. 
The same beam theory, combined with mode superposition technique is used 
in [4]. Geometrically nonlinear beam theories are used in [5,6] for the 
modeling of a single-link and multi-link flexible robotic manipulators, 
correspondingly. 

Taking into account the fact that robotic arms are usually not very thin and that 
the transverse shear could play an important role for dynamically loaded 
structures [7] the application of the Bernouli-Euiler beam theory could lead to 
a discrepancy between the robotic arm behavior and that one described by the 
model. 


1275 


The aim of this work is to model the dynamic behavior of a single link flexible 
robotic arm employing the Timoshenko beam theory, which considers the 
transverse shear and rotary inertia. The arm is subjected to a dynamic loading. 
As in [3], the viscous friction is included into the model and slip-stick 
boundary conditions of the rotating hub are introduced. Besides that, the 
possibility of the rise of a contact interaction between the robotic arm and the 
stop (limiting support) is included into the model. The beam stress state is 
checked for plastic yielding during the whole process of deformation and the 
plastic strains (if they arise) are taken into account in the model. The 
numerical results are provided in order to clarify the influence of the different 
parameters of the model on the response of the robotic arm. 

2. BASIC EQUATIONS 
2.1. Formulation of the problem 

The robotic arm - flexible beam is attached to a rotor that has friction and 
inertia. The beam is considered to be clamped to a rotating hub and its motion 
consists of two components: “rigid-body” component and a component 
describing the elastic deflection of the beam (see Figure 1). The motion of the 
flexible beam is accomplished in the horizontal plane and gravity is assumed 
to be negligible. 



Figure 1. Model of one-link flexible manipulator. 


1276 




Tip of the beam (with attached tip-mass) is subjected to an impulse loading. 
Stick-slip boundary conditions due to Coulomb friction of the hub are 
introduced when describing the motion of the beam. In other words, if the 
bending moment, about the hub axis, due to the impact is lower than the static 
friction torque threshold then the hub is considered clamped and the beam 
elastic motion is considered only. When the bending moment exceeds friction 
torque threshold this boundary condition is removed, allowing rotation of the 
hub and the arm. When the hub speed and kinetic energy of the beam become 
again beneath the torque threshold, the hub clamps again. 

The possibility of the rise of a contact interaction between the robotic and the 
stop is envisaged. In this case, if the hub angle exceeds the limit value, the 
robotic arm clamps and a part of the beam goes in a contact with the stop, 
which is modeled as an elastic foundation of a Vinkler type. 

In view of the fact that the impact loading and contact interactions are 
included in the investigations, it is expedient to be considered the rise of 
plastic strains in the beam. 


2.2. Deriving the equations of motion. 


The total kinetic and potential energy of the rotating hub with the attached 
beam (described by the Timoshenko beam theory) and a lumped mass at it’s 
tip can be expressed as follow: 

Ek =^|pA[u(x,t)] dx + |j„[e(t)]^+tMT[u(l,t)fjEl[(()(x,t)] dx 




EIl^l +kGA’ 
dx. 




V dx 


dx 


(1 a,b) 


In these equation u(x,t) is the total displacement 
u(x,t) = w(x,t)+x0(t) (2) 

and w is the transverse displacement of the beam, (p is the angular rotation, 
9 is the hub angle, E is the Young modulus, G is the shear modulus. A: is a 
shear correction factor, p is the material density, Jh is the inertia moment of 
the hub, A=b*h is the beam cross-section area, h is the thickness, b is the 
width, I=bh^/12,1 is the length of the beam, Mt is the tip mass, t is the time. 


1277 



Denoting the work of external forces (applied actuating torque T(t) and the 
beam loading p(x,t))by 


W=T(t)e(t)+ \p{x,t)w{,x,t)dx 

0 

the Hamilton's principle can be applied: 
j5(E,-Ep)dt+ j5Wdt = 0 


(3) 

(4) 


Substituting eqns (l)-(3) into eqn (4) after integrating and including damping 
of the beam material, the viscous friction of the hub and the reaction force of 
the elastic foundation the following equations of motion can be obtained: 


. r . X. i 2 x de , ,5^w(l,t) , 

(J + Jh + M-j -1 ^ j ^2 ^ ^ ^ ^2 


i 

JpAx 


a^w(x,t) 

dt^ 


dx = T(t) 


EI^-^ + kGA 
dx^ 


dw 


dx 


dw 


.d^cp 


-9 “ <^2 - —2 ~ 0<x</, t>0 (5a-c) 


dt 


dt 


kGA 






+ x- 


a^0 

'St" 


-R(x,t) = p(x,t) 


The boundary conditions are: 
w(0,t)=(p(0,t)=0 


dcpgt) Q 
dx 


kGA 


dw(l,t) 

dx 



d^w(lt) 

dt^ 



( 6 a-c) 


and the initial conditions are: 

wfx, 0) =w^(x), w(x, 0 ) = w° (x), p(x,0)=(f^(x), (p(x, 0 ) = 9 ° (x) 

e(t) = e(t) = o, t<t,,p 

or 9(0 = 

when | 9 (t)| and kinetic energy of the beam falls simultaneously under the 
Coulomb friction thresholds. 

In eqns (5) the viscous friction coefficient of the hub is denoted by di, d2 and 
d^ are damping coefficients of the beam material, J is the moment of inertia of 
the beam about the motor axis and R(x,t) is the reaction force of the stop 
disposed from Xj to X 2 (xj<X 2 <l) and modeled as an elastic foundation with 
Vinkier constant r: 


(7a-d) 
(7 e,f) 
(7e)’ 


1278 


R(x,t) = 


r[w(x, t) - (x)] for x j < x < x^; |0l >' 

0 for 0 <x<x,,X 2 <x</; iBi < 


The time when the bending moment about the hub axis exceeds the friction 
torque threshold is denoted by tsUp^ When t>tsiip this condition is removed 
(allowing rotation of the hub and the beam) until the moment when the beam 
clamps again. 

2.3. Elastic-plastic relationships 

The beam stress-strain state is usually expressed in terms of generalized 
stresses and strains which are function of x coordinate only. As a unique 
yield criterion in terms of moments and the transverse shear force does not 
exist according to Drucker [8], the beam cross-section is divided into layers 
and for each of them the stress state has to be checked for yielding. 

The relation between the stress vector S = and the strain vector 

s = |-z ^I ’ generally presented as 


S=[D]8, 

where in the case of an elastic material [D] = [D^] = 


is the elastic 


matrix and f(z) is a function describing the distribution of the shear strains 
along the thickness. 

On the basis of the von Mises yield criterion, the yield surface is expressed 


After yielding during infinitesimal increment of the stresses, the changes of 
strains are assumed to be divisible into elastic and plastic parts 


As = As^ + As^ 


where 


As 


1279 


By using eqns (10), (11) and the associated flow rule [9], following Yamada 
and others [10], the following explicit relationship between the increments of 
stresses and strains is obtained 


. AS = [d‘'’]A8 

where [-0'^] is the elastic-plastic matrix: 



HY\d^] 

LjaLiVi 

r^r 

1 J 1 J L j\asj 


1 - 

C/2 

1— 

108] 


In this equation H is a function of the hardening parameter. For ideal plasticity 
H is equal to zero, while for a wholly elastic material H -> co . 

3. Use of mode superposition. 

3.1. Rearrangement of the equation of motion. 

Let the total time interval T on which the dynamic behavior of the structures is 
investigated, be divided into sequence of time increments y+J. 

In the numerical calculations the following dimensionless variables are used: 
x = xl U w = w//, t =tl /c c = / p 

and then omitting the bars, and after some algebra, the governing equations 
can be written in the following form: 


d'e de 

- T- + C, - 

d t^ ' d t 


--cJ w(l,t)- f(p(x,t)dx =C 3 T(t)+ fxp(x,t)dx 
tv 0 /Vo / 


09 0^9 -I c/w I 


(15 a-c) 


0^W 01V 0^iv 09 

dt^ ^ dt 10x ^ dx 


= -p~G{-G^ 


where a=12P/h^ , p~kG/E, p=pI/(EA), c,= d^l/(cJ}{), Arp/(JHp) C 3 = 
P/(c^J}^ ), C4~d49/EI, C 3 =djl/EA, =5/6. The nonlinear force due to the 

reaction of the foundation is denoted by G^ = R(x,t).l/EA and Gf and G 2 are 
the components of the so-called non-linear force vector Gp( .G^} which is 
due to the inelastic strains. It has the presentation (see [7]): 


1280 




3.2 Mode superposition method 

The l.h.s. of eqns (15 b,c) is a linear form and therefore the mode 
superposition method can be used for its solution. As the eigen frequencies 
and the normal modes of vibrations of an elastic beam do not correspond to the 
real nonlinear system, these modes are called "pseudo-normal" modes. 

Thus, the generalized displacements vector v = |a”^ (p, w| is expanded as a 
sum of the product of the vectors of pseudo-normal modes v^ and the time 
dependent functions q„(t) as 

( 17 ) 

n 

The analytically obtained eigen functions of the elastic Timoshenko beam are 
chosen as basis functions (see APPENDIX). When the tip mass is attached to 
the beam the eigen functions of the system used in the mode superposition 
method are preliminary orthogonalized by standard orthogonalization 
procedure as it is mentioned in the APPENDIX. 

Substituting eqn (17) into (15 b,c), multiplying by v^(x), integrating the 
product over the beam length, invoking the orthogonality condition (see (A9)) 
and assuming a proportional damping for the beam material 

|(c 4 (pf, +c^wl)rdr the following system of ordinary differential 

equations (ODE) for 0(t) and qn(t) is obtained: 


^(f) + c, (9(0 = Cj 2] [«'»(1) - ]?» 


^ » 


+ C3(r(0 + P(0) 


(18a,b) 


9„(o+2f„ffl„9„(o+®,k,(o=-‘y„ -'f„&(o+&!’ 

In this equations 

1 

11 V f 

Y„ = J(p„ 6„ = = jxw,Xx)ck, P(t)= \xp{x,t) dx , 


1 1 

g;(t)= |G'’(x,t)v„(x)dx, g',(t)= jG'(x.t)w„(x)dx, co„ are the eigen 

0 0 

frequencies of the elastic clamped beam and are the modal damping 
parameters. 

The initial conditions defined by eqns (7 a-d) are transformed also in terms of 
q„(0), and ?„(0) 

9 „( 0 ) = 9 „“, 4 „( 0 ) = 4 °, 

q°„ = J(w°w„ t-a'VVjdx, (j“„ = j(w"w„+a-'(p°(p„)dx, (19a-d) 

0 0 

The obtained system of nonlinear ordinary differential equations is a stiff one 
and it is solved numerically by the backward differential formula method, also 
called the Gear's method [11]. 

The rise of plastic strains is taking into account by using an iterative procedure 
based on the "initial stresses" numerical approach [7]. 


4. RESULTS AND DISCUSSION 

Numerical results were performed for the robotic-arm with the same material 
and geometrical characteristic as these given in [3] in order to make some 
comparisons. Model parameters are: E=6.5xl0 Pa ,v=0.2, Cp=2.6xl0 Pa, 
1=0.7652, b=0.00642 m, h=0.016 m, p=2590 kgW , Mt=0,153 kg, Jh=0.285 
kgm^ (Jh is not defined in [3].) 

The aim of the computations is to show and clarify the influence of the elastic 
or elastic-plastic deformation on the motion of the robotic-arm, to demonstrate 
the effect of the hub friction, slip-stick boundary conditions and the contact 
interaction between the beam and the stop. 


1282 


Only impact loading on the beam is considered in this work, i.e. dynamic load 
p(x,t) and applied torque T(t) are equal to zero. The impact loading is 
expressed as an initial velocity applied to the tip of the beam 0.95 < x < 1. 

Nine modes are used in expansion (15) but the results obtained with number of 
modes greater than nine are practically indistinguishable from these shown 
here. 

For all calculations the material damping is equal to 8% of the critical 
damping. 

The results for the rotation of the hub of the flexible manipulator with an 
attached mass at its tip -1 and without an attached mass -2 are shown in Figure 
2. The hub friction is not considered. The beam is subjected to an impact 
loading with initial velocity =-1.95 m/s. As can be expected, the hub angle 
increases much faster in the case of the beam with an attached mass. The 
corresponding beam deflections are presented in Figure 3. The results obtained 
are very close to these obtained in [3] (Fig. 7 and 8 in [3]). The frequencies of 
forced vibrations obtained in [3], however, correspond to the beam without an 
attached mass. 


0, rad 



-4.00 I-^^^^-1- ^^i^ 

0,00 l.OO 2.00 3.00 4.00 5.00 

t, sec 

Figure 2. Hub response without 
viscous friction. 1 - beam with an 
attached mass; 2- without an attached 
mass 


w, m 



Figure 3. Deflection of the tip of the 
beam without viscous friction of the 
hub. 1 - beam with an attached mass; 
2 - without an attached mass 


1283 


The influence of the hub friction on the flexible manipulator response can be 
seen in Figure 4. The viscous friction is set di=0.1 Nms, the static Coulomb 
friction threshold is equal to 0.06 Nm and three cases of the hub slip-stick 

threshold are tested: |e| stick = 0-005 rad.s'^ - 1, |0| stick = 0.0085 rad.s'^ , 

stick = 0.01 rad.s'^ For this initial velocity (w° =-1.95m/s) the hub slips 

very fast from the initial clamped state and the beam begins to rotate. As can 
be expected, the consideration of the viscous friction of the hub leads to a 
decrease of the angle of rotation of the beam and changes the linear variation 
of 0 with time. The results show also that the value of the hub speed 

threshold |0| stick exercises an essential influence on the motion of the 
rotating system. 

When |e| =0.005 rad.s' the 

beam sticks at t = 2.602 s after that 
the hub periodically slips and sticks 
which also leads to damping of the 

motion. When joj stick “ 0.0085 

rad.s'^ the start of sticking occurs at 
t = 1.7207 s and after t=2.417 s the 
hub clamps with short interruptions 
till t=3.4 s when due to the elastic 
vibrations it snaps in the direction 
opposite to w°» clamps again at 
3.679s, slip at 4.5s, and finally 

clamps at t=4.5s. When joj stick 
=0.0085 rad.s"' the sticking begins 
at t=1.0525s and very fast (at t«2 s) 
the beam clamps with 0=-O.587 rad. 

In order to observe the occurrence of the plastic deformation the beam was 
subjected to impulse loading having larger values of initial velocities. In 
addition, the contact interaction between the beam and the stop was 
considered. 

The beam-tip deflection in the presence of a contact with the stop disposed at 
X €[0.16, 0.263] and initial velocity w° =-15.95 m/s is shown in Fig. 5. 


0,rad 



di=0.1Nms. 1- |0| stick”0-005rad.s"', 

2 - |0| stick = 0-0085 rad s"' ; 

3- |0| stick =0-0is-' 


1284 


In order to reduce the computational time the limit value of 0 was chosen 
e^j^p=0.0025 rad. When this value was reached the problem was automatically 
resolved with new initial conditions 0=0, w°=w(Xstop,tstop), etc. 

As can be seen, the presence of the contact interaction during the process of 
motion of the beam due to the elastic support for x e[0.16, 0.263] leads to a 
decrease of the amplitudes of vibrations in the direction of the stop in 
comparison with the amplitudes in the opposite direction. The variation of the 
beam displacements along the beam length for the first 0.8 s of motion is 
shown in Fig. 6. It must be noted that in this case of loading a plastic yielding 
occurs. It is assumed that beam material is characterized by an isotropic linear 
strain hardening and H=0.5. The plastic yielding occurs simultaneously with 
the contact interaction at t=0.01366 s at the clamped end of the beam. At 
t=0.0186 s the plastic zone spreads to x = 0.158 and at t=0.08767 if covers the 
length to X =0.31. The last points that yields are x =0.55, 0.61 at t=l. 119 s. 
Seven layers along the beam 
thickness, symmetrically disposed 
about the beam axis was checked for 
yielding (N2=7) but the plastic zone 
has reached the second and 6th 
layers only at the clamped end of the 
beam (x =0). In all other point along 
beam length the plastic yielding 
occurs only at the upper and lower 
surface of the beam. The plastic 
strains are small and the response of 
the beam is not very different from 
the wholly elastic response. 

Nevertheless, the appearance of such 
kind of plastic deformations in the 
structures used for the precise 
operations must be taken into 
account in the manipulator self 
calibration procedure. 


w,m 

0.40 

0.30 

0.20 

0,10 

0.00 

-O.IO 

- 0.20 


-0.30 




0.00 


1.00 


2.00 


3.00 


4.00 5.00 

t, sec. 

Figure 5. Deflection of the tip of the 
beam with time in the case of a 
contact with the stop. r=5.5xl0^ Pa 


5. CONCLUSIONS 

In this work a model describing dynamic behavior of a deformable beam 
attached to a rotating hub that has friction and inertia is developed. The 
Timoshenko beam theory is used to model the elastic deformation of the beam. 


1285 









Figure 6, Variation of the elastic-plastic beam displacement along the beam 
length with time in the case of a contact with the stop. 

The slip-stick boundary conditions are also incorporated into the model 
The possibility of the rise of undesired plastic deformations in the case of a 
high velocity impact on the clamped robotic arm, or in the case of a contact 
with limiting support (stop) is included into the model. 

The analytically obtained eigen functions of the elastic Timoshenko beam 
vibrations are used to transform the partial differential equations into a set of 
ODE by using the mode superposition method. This approach minimizes the 
number of ODE which have to be solved in comparison with another 
numerical discretization techniques (finite elements or finite difference 
methods). 

The results obtained show the essential influence of elasticity on the robotic- 
arm motion. 

The model will be used to synthesise a control of one link flexible 
manipulators and for a self calibration procedure when plastic deformation 
would occur. 

Acknowledgments 

The authors gratefully acknowledge the financial support from EC Copernicus 
Program under the Project ROQUAL CIPA CT 94 0109. 

The first author wishes to thank the National Research Fund for the partial 
financial support on this study through Contract MM-517/95. 


1286 



REFERENCES 


1. Bayo, E. A finite-element approach to control the end-point motion of a 

single -link flexible Robert. J. Robotic System, 1987, 4„ 63-75 

2. Bayo, E. and Moulin, H., An efficient computation of the inverse dynamics 

of flexible manipulators in the time domain. IEEE Proc Int. Conf. on 
Robotics and Automations, 1989, 710-15. 

3. Chapnik, B.V., Heppler, G.R., and Aplevich, J.D. Modeling impact on a 

one-link flexible robotic arm. IEEE Transaction on Robotics and 
Automation, 1991,7,479-88. 

4. Liu, L. and Hac, A., Optimal control of a single link flexible manipulator. 

Advances in Robotics, Mechatronics, and Haptic Interfaces, 1993, DCS- 
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5. Wen, J.T., Repko, M. and Buche, R., Modeling and control of a rotating 

flexible beam on a translatable base. Dynamics of Flexible Multibody 
Systems: Theory and Experiment, 1992, DCS-37, 39-45. 

6. Sharan, A.M. and Karla, P., Dynamic Response of robotic manipulators 
using modal analysis. Meek Mach. Theory, 1994, 29, 1233-49. 

7. Manoach, E. and Karagiozova, D. Dynamic response of thick elastic-plastic 

beams. International Journal of Mechanical Sciences, 1993, 35, 909-19 

8. Drucker, D.C. Effect of shear on plastic bending of beams. J. of Applied 

Mechanics, 1956,23, 515-21 

9. Hill, R. Mathematical Theory of Plasticity, 1950, Oxford University Press, 
London. 

10. Yamada, Y., Yoshimura, N. and Sakurai T., Plastic stress-strain matrix and 
its application for the solution of elastic-plastic problems by the finite 
elements, Int. J. of Mechanical Sciences , 1968,10, 343-54 

11. Gear, C.W., Numerical initial value problem in ordinary differential 
equations, 1971, Prentice-Hall, Englewood Cliffs, NJ. 

12. Abramovich, H. Elishakoff, 1. Influence of shear deformation and rotary 
inertia on vibration frequencies via Love’s equations. J. Sound Vibr., 1990, 
137, 516-22. 


1287 


APPENDIX: NORMAL MODES OF FREE VIBRATIONS OF A 
CLAMPED TIMOSHENKO BEAM WITH AN ATTACHED MASS, 


Equations (5 b,c) can be decoupled, transforming them into two fourth order 
equations [12] as regards cp and w. 

Solving this equations (with p=0 and R=0) and using the boundary conditions 
(6) (with 0=0), the equations of the frequencies and forms of vibrations of the 
beam are obtained. 

Introducing following denotations 

1/2 (A1 a-e) 

^2„={®,^(l + P) + [®:a + P)^+4(B,^(a-pco,^,)]''^} /2, 

/l„ =(4 +CoJP)/5,„, /2„ /3„ =(4-®SP)''-S!». 

the frequencies of free vibrations are determined as roots of equation: 


a) In the case >0 i.e. co < a / p the frequencies equation is: 

^11^22 ”^12^21 “ ^ > (-^2) 
where 

^11 ~/ 2 n‘^ 2 n ^12 ~ ■^/ 2 «‘^ 2 « ®^^(‘^ 2 «) 

h\ = -f\nM.Sxn ) “ /2„ sin(52„ ) + s^„sh(si, ) + 52„ sin(52„ ) + , . . 

(A3 a-d) 

M‘^lnCh(^lJ + ‘^2« cos(,y2J] 


*22 = /2j=OS(^2«)-cll('SlJ] + 4^S,„ch(i,„)-52„ COS(52„) + 

J\n 

+ sl„ sin(i 2 „)] 

Jin 


and the modes of vibrations are: 

w„W = -S„ 

1 

' b 


^(ch(i,„x) - cos(j2„x)) + sin(i2„x) - ^sh(i|„x) 

J\» 




L^i 


(A4 a,b) 

(/,„sh(,s,„x) + sin(,y 2 «^)) + fi,, icos{s^„x) - ch(s,„x) 


b) In the case 5 ^,, <0 i.e. > a / P the eigen frequency equation (A2) has 
the following presentation: 


1288 


b\\ = + /2„i2„ C0S(J2„) 

bn = ?i„/2nSin(?i„) - /2„i2n sin(j2„) 

621 = /3„sin(ii„) - /2„ sin(52„) -?i„sin(Ji„) + S2„ sinC^jJ - 
>.[J,^„COS( j,„ ) - sl„ C 0 S(i 2 „)] 

*22 = /2„[cOS(S2„) - COS(J,„)] - ^ S,„COS(?,„) + S 2 „ COS{s-^„) - 

J\n 

- sl„ sin(52„)] 

7 In 

sh = -5?„ 


and the of vibrations are: 

w„(x) = 


f b 

sin(j2„2c) - ^sin(?,„x) - ^(cosCij,,*) - cos(j,„x)) 
/i„ *11 


'P„W = -S» 


(A 5 a-e) 


(A6a-b) 


/j,, (cos(i2„x) - cos(^|„x)) - ^(/3„sin(i,„x) - f^„ sin(jj„x)) 

0,, 


When a mass is not attached at the beam tip the following orthogonality 
condition is fulfilled: 

1 


p fO, n^ m; 

(a _ 

J [1, n - m. 


(A 7 ) 


and when an attached mass is considered the modes are orthogonalized by 
standard orthogonalization procedure. 

The constants are obtained from condition (A 7 ). 


1289 




1290 


Wave Reflection and Transmission 
in an Axially Strained, Rotating Timoshenko Shaft 


B. Kang ‘ and C. A. Tan ^ 
Department of Mechanical Engineering 
Wayne State University 
Detroit, Michigan 48202, U. S. A. 


Abstract 

In this paper, the wave reflection and transmission characteristics of an axially strained, 
rotating Timoshenko shaft under general support and boundary conditions, and with geometric 
discontinuities are examined. The static axial deformation due to an axial force is also included 
in the model. The reflection and transmission matrices for incident waves upon these point 
supports and discontinuities are derived. These matrices are combined, with the aid of the 
transfer matrix method, to provide a concise and systematic approach for the free vibration 
analysis of multi-span rotating shafts with general boundary conditions. Results on the wave 
reflection and transmission coefficients are presented for both the Timoshenko and the simple 
Euler-Bemoulli models to investigate the effects of the axial strain, shaft rotation speed, shear 
and rotary inertia. 


‘ Graduate Research Assistant. Tel: +1-313-577-6823, Fax: +1-313-577-8789. E-mail: kang@feedback.eng.wayne.edu 
’ Associate professor (Corresponding Author),Tel: +1-313-577-3888, Fax:+1-313.577-8789. E-mail: tan@tan.eng.wayne.edu 

Submitted to: Sixth International Conference on Recent Advances in Structural Dynamics, Institute of Sound and Vibration, 
Southampton, England, July, 1997 


1291 


A. 

NOMENCLATURE 

Area of shaft cross section [m^] 

do 

Diameter of shaft cross section [m] 

C 

Generalized coordinate of an incident wave [m] 

Cdt {Ct) 

Translational damping coefficient [N-sec/m] 

Cdr (Cr) 

Rotational damping coefficient [N-m-sec/rad] 

Co 

Bar velocity [m/sec] 

Cj 

Shear velocity [m/sec] 

D 

Generalized coordinate of a transmitted wave [m] 

E,G 

Young’s and shear modulus [NW], respectively 

I 

Lateral moment of inertia of shaft [m'^] 

Jm (Jm) 

mass moment of inertia of a rotor mass [kg-m"^] 

K 

Timoshenko shear coefficient 

Knikr) 

Rotational spring [N/rad] 

Kt (kd 

Translational spring [N/m] 

t 

Length of shaft [m] 

M{m) 

Mass of rotor [kg] 

P 

Axial force [N] 

rij, ti] 

Reflection and transmission coefficients, respectively, i - 1 positive traveling 

U{u) 

wave; i = 2 negative traveling wave; jr = 1 propagating wave for Cases 11 and 
/V; 7 = 2 attenuating wave for Cases II and IV. Both 7 =1, 2 for propagating 
wave for Case I 

Transverse displacement [m] 

X-Y-Zix-y-z) 

Reference frame coordinates [m] 

a 

{K-G)IE 


Rotation parameter, see Eqn. (Id) 

£ 

Pl{E-A), axial strain 

e' 

Non-dimensional axial load parameter, see Eqn. (13b) 

T, f (r, y) 

Wavenumber [m‘‘] 

nAi) 

See Eqns. (20a, b), (22a, b) and (24a, b) 

p 

Mass density of shaft [kgW] 

G 

Diameter ratio between two shaft elements 

57, (ft)) 

System natural frequency for Timoshenko model [rad/sec] 

a 

System natural frequency for Euler-Bernoulli model [rad/sec] 

Q. 

Rotation speed of shaft [rad/sec] 

W (¥) 

Bending angle of the shaft cross-section [rad] 

subscript L, r 

The left and right side of a discontinuity, respectively. 

superscript -,+ 

Negative and positive traveling waves, respectively, when used in C and D. 

Note: Symbols in 

Otherwise denotes quantities on the left and right side of a discontinuity, 
respectively 

parenthesis are the corresponding non-dimensional parameters. 


1292 



1. INTRODUCTION 


The vibrations of elastic structures such as strings, beams, and plates can be described in 
terms of waves propagating and attenuating in waveguides. Although the subject of wave 
motions has been considered much more extensively in the field of acoustics in fluids and solids 
than mechanical vibrations of elastic structures, wave analysis techniques have been employed to 
reveal important, physical characteristics associated with vibrations of structures. One advantage 
of the wave technique is its compact and systematic approach to analyze complex structures such 
as trusses, aircraft panels with periodic supports, and beams on multiple supports [1]. Previous 
works based on wave propagation techniques have been well documented in several books [2-4], 
Recently, Mead [5] applied the phase-closure principle to determine the natural frequencies of 
Euler-Bemoulli beam models. A systematic approach including both the propagating and near- 
field waves was employed to study the free vibrations of Euler-Bemoulli beams [6]. 

High speed rotating shafts are commonly employed in precision manufacturing and power 
transmission. Despite the usefulness of the wave propagation method in structural vibrations, 
applications of this technique to study the dynamics and vibrations of a flexible shaft rotating 
about its longitudinal axis have seldom been considered. The purpose of this paper is to examine 
the wave reflection and transmission [6] in an axially strained, rotating Timoshenko shaft under 
various support and boundary conditions. The effect of the axial load is included by considering 
the axial static deformations in the equations of motion. This paper is a sequel to another paper 
in which the authors discuss the basic wave motions in the infinitely long shaft model [7]. 

Although there have been numerous studies on the dynamics and vibration of rotating shafts, 
none has examined the effects of axial strains (which cannot be neglected in many applications) 
on the vibration characteristics of a Timoshenko shaft under multiple supports. Modal analysis 
technique has been applied to study the vibration of a rotating Timoshenko shaft with general 
boundary conditions [8, 9], and subject to a moving load [10]. Recently, the distributed transfer 
function method was applied to a rotating shaft system with multiple, geometric discontinuities 
[11], The wave propagation in a rotating Timoshenko shaft was considered in Ref. [12]. Other 
major works on the dynamics of rotating shafts have been well documented in Refs. [13-15]. 

This manuscript is organized as follows. Governing equations of motion [16] and basic wave 
solutions for the Timoshenko shaft are outlined in Section 2. Each wave solution consists of four 
wave components: positive and negative, propagating and attenuating waves. In Section 3, the 
wave reflection and transmission matrices are derived for the shaft under various point supports 
and boundary conditions. The supports may include translational and rotational springs and 
dampers, and rotor mass. Results are presented for both the Timoshenko and the simple Euler- 
Bemoulli models to assess the effects of axial strain, shaft rotation, shear and rotary inertia. The 
wave propagation across a shaft with geometric discontinuities such as a change in the cross- 
section is examined in Section 4, and the wave reflection at a boundary with arbitrary support 
conditions is considered in Section 5. 

With the wave reflection and transmission matrices as the main analytical tools, it is shown in 
Section 6 how to apply the current results together with the transfer matrix method to analyze the 


1293 


free vibration of a rotating, multi-span Timoshenko shaft system in a systematic manner. The 
proposed approach is then demonstrated by considering the free vibration of a two-span beam 
with an intermediate support. 


2. FOKMULATION AND WAVE SOLUTIONS 


Consider a rotating shaft subjected to axial loads and with multiple intermediate supports and 
arbitrary boundary conditions, as shown in Fig. 1. Including the effects of rotary inertia, shear 
deformations, and axial deformations due to the axial loads, the uncoupled equations of motion 
governing the transverse displacement u and the slope \}/ due to bending can be derived in the 
following non-dimensional form 


.. . d'^u „ d^u 

£ 

+16a(l + £)(l-i-e-)^r^ = 0 , 

a dr 


U lA £ . d~U 

■2ri3^ + a—-16£{1 + £--)^^ 




a dz'^ 


-l-16a(l-l-£)(l + £-—= 0 , 
a dr 


u = — z = — r = — T = ^ ■ 
a.' a.' T. ^ \ KG 



Figure 1. A rotating Timoshenko shaft model subject to axial loads and with general boundary conditions. 


1294 


(Id) 


a 


KG 
E ’ 



E 



Note that u and y/are the measurements in the complex plane, that is u=ux+iuy and \i/=\}fx+iYr 
E denotes the Young’s modulus, p the mass density, As the area of the cross section, ao the 
diameter of shaft, K the Timoshenko shear coefficient, G the shear modulus and Q the constant 
angular velocity of the shaft. Details of deriving these equations of motion are found in Ref. 
[16]. 

Assuming and substituting the following wave solutions into Eqns. (la) and (lb) 

= ( 2 a) 

= ( 2 b) 

and defining the non-dimensionalized wavenumber 7 and system natural frequency w gives the 
frequency equation, Eqn. (3a); see Ref. [7], 


y = r^o . 


_ CO a, 


KG . 


(j) = -a. = -is known as the shear velocity). 


where, 


7 “* -A 7 " 4-5 = 0 , 


A = (1 -i- a)a) ^ - 2j3 cu “ 16£ (1 + e-), 


5 = a ot)^ - 2j3 u) - 16a (14-£)(1 4 -e - •^) 


The four roots of Eqn. (3a) are 


7 = ±-^[a ± -x/a^ -45 . 


( 2 c) 

( 2 d) 

(3a) 

(3b) 

(3c) 

(4) 


In general, 7 is complex. Let (o be real. It can be shown that, with a > 0 and £ the axial strain 

of the elastic solid, the discriminant A^ -45 is positive semi-definite for most engineering 
applications. Hence, it is possible to classify the wave solutions into four distinct cases. Note 
that one may study the wave propagation by considering only a single general form of the wave 
solution. However, the classification procedure identifies the coupled modes of vibration of the 


1295 


Timoshenko shaft model and provides a better understanding on how each wave solution governs 
the wave motions [7]. Based on the algebraic relationships between A and B, the four valid wave 


solutions are obtained as follows. 

Case / (A > 0 and B > 0); 

«(z, t) = + C-(5a) 

V(z, t) = (CJ.e-'*’" + )e® (5b) 

Case !I{A>0 andB<0): 

«(z, t) = (Ce-"''' + + C:^e-^'-‘ + (6a) 

yz(z,t) = (c;,^-'^" - c;,/" + + 0 ;^/=- )£® (6b) 

CaseIIIiA<0mdB>0): 

uiz,t) = (C>-''' + + C:,«")«'“ (7a) 

Vr(z,0 = (C;,a-" + C;,ef>' + + C;je'=')«® (7b) 

Coje/V(A<0 andB<0): 

k(zA) = (C,>-^'^ +C,>’‘'- + C>-'''=‘ + C,V'’'*')«'® (8a) 

V/(z,«) = (C;,e-f- + (8b) 

where, 

f, =^(|A| + V-5’-4|B|f, r. (9^. W 

r, = + 4|B| + |A|)^, r, + 4|B| - |A|j', (9c,d) 


and the coefficients C* and C" denote positive- and negative-travelling waves from the origin of 
disturbance, respectively. Important remarks on the basic wave propagation characteristics are 
summarized from [7]. First, the wave solution of Case III does not exist in the real frequency 
space since this type of solution represents a situation in which none of the wave components can 
propagate along the waveguide. Therefore the study of Case III is excluded in the present paper. 
Second, the vibrating motion of the shaft model in Case I is predominately pure shear [17] which 


1296 


is unique for the Timoshenko shaft model, while in Case U and Case IV the flexural mode and 
the simple shearing mode, which are corrected by including the rotary inertia and shearing effects 
in the formulation, dominate. Third, when the shaft rotates at a very high speed and/or the shaft 
is axially strained by tensile loads, the wave solution of Case IV governs the vibrating motion of 
the shaft model in the low frequency range. 


For comparison, the parameters A and B in the simple Euler-Bernoulli beam model are 



( 10 a) 


( 10 b) 


where, the non-dimensionalized natural frequency 6 ) is defined as 



is known as the bar velocity). 


( 10 c) 


Note that, because B is negative, wave solutions of Case I and Case III do not exist. 

In general the displacement and the rotation of an infinitesimal shaft element consist of four 
wave components as shown by Eqns. (5a-8b). Once the displacement and the bending slope are 
known, the moment M and shear force V at a cross section can be determined from 


M = EI^, 
dz 

(ll) 

(du ^ 

y =J. 

(12) 

Moreover, the kinematic relationship between the transverse displacement and the slope due to 
bending is 

S^u 3^u . ,3 r 

3r~ 3z^ 3z ' 

(I3a) 

where e' denotes the effects of the axial force and is defined as 


e = l + e —. 

a 

(I3b) 


1297 



3. WAVE REFLECTION AND TRANSMISSION AT SUPPORTS 


When a wave is incident upon a discontinuity, it is transmitted and reflected at different rates 
depending on the properties of the discontinuity. Consider a rotating Timoshenko shaft model 
supported at ^ = 0; see Fig. 2. The support simulates a bearing modelled by linear, translational 
and rotational springs, dampers, and a rotor mass which typically represents a gear transmitting a 
torque. Based on Eqns. (5a-8b), group the four wave components into 2x1 vectors of positive¬ 
travelling waves and negative-travelling waves C", i. e.. 



Recall that, depending on the system parameters, the rotating Timoshenko shaft model has four 
(practically three) different wave solutions in the entire frequency region as described in Eqns. 
(5a-8b). Thus C^ and C, in the above expression do not always correspond to propagating and 

attenuating wave components, respectively. When a set of positive-travelling waves is 
incident upon the support, it gives rise to a set of reflected waves C~ and transmitted waves . 
These waves are related by 


C'=rC" 

(15) 

D" = tC", 

(16) 


where r and t are the 2x2 reflection and transmission matrices respectively and are expressed as 


r = 

'^u 

^ 12 * 

(17) 


/ 2 ! 

'' 22 . 



’^11 

^12 


t = 


(18) 


/ 2I 

^22 _ 


From Eqns. (5a-8b), suppressing term and excluding Case III, the displacements u~ and 
and the bending slopes y/~and y/'^ at the left and right of z = 0, respectively, can be expressed 
in terms of the wave amplitudes of the displacement. For convenience, the over-bar (•) on the 
wavenumbers is dropped hereafter. 


Case 1 {A >0 andB>0); 

«-(z) = C,+ c;,£'^==, (19a) 

W'U) = -n,C:,e-‘^“ -77,c;,+T7,C>-'>''-(19b) 


1298 


M.Jm 



Figure 2. Wave motion at a general support (the disk may be considered as a gear transmitting a torque). 


where, 

_ 72-®' 

T/i — / » ~ / • 

r,£ 72^ 

// (A > 0 and B < 0): 

«-(z) = c;,+ C'e"''' + , 

i^-(z) = 7,,C,>-'"■= -77,C,-^r''''= +)7,C*e-''“ -TJ,C,;;e"=S 


k*(z) = A>“’'" + AV’‘. 


where 


= 


r,£' 


> Ba “ 




CaselV{A<0a.ndB<0): 


(19c) 

(19d) 

(20a, b) 

(21a) 

(21b) 

(21c) 

(21d) 

(22a, b) 


1299 



the following set of matrix equations can be established for each Case. 
Case / (A > 0 and 5 > 0): 


n n 

r 1 

1 1 

. ri n 

c^ + 

rC^=: 

L^i 12 ] 


-^2j 



-im, Ic+r IrC" 

KYi-tIi) i(r2“7?2)J [-KYi-Vi) -i(.Y 2 -Tl 2 )\ 


(28a) 


ri,(k,-J„,co^) + iTi,{c,0)~r,) 0)") +177272) ^28b) 

ik^~mO)-) + i{c,co + r^-r]^) (fc,-mtt>") + z’(c,® + 72- 772 ) J 


1300 


Case // (A > 0 and S < 0): 


1 

.^1 


1 




1 


L^i 


1 

^2 


tc^ 


1 

1 

n 


-ir,T7, 


[i(r,-T7i) r2-j772_ 

C -r 

-^r,-77,) 

~ (r, - it ],)_ 


(29a) 


r},(k^-J„^0)-) + irij (c^co-r^) 
(k, -mco^) + i(c,Q) + r,-T],) 


Tl2(k,-J„,co^-r2) + iV2CrO} 

{kj -m(0^ +r2)+/(c,Ct}- 772) 


tc", 


(29b) 


Case /y(A<0and5<0); 


'1 11 


■ 1 1 ■ 

rC" = 

' 1 r 

Jli Hi. 


-^2 -^ 1 . 


jii ^ 1 - 


■-1X2772 -£t, 77 , ■ 

C'* 

-1X2772 


KT.-ni) r,-i 77 ,_ 

-r 

— i(r 2 — 772) 

-(r, - in,). 


(30a) 


T]2ik^-J„^(0'") + iT]^{c,C0-y2) 
{k, -m(D‘) + i(c,aj + r2 - 772 ) 


riiik,-J„co^-y,) + iT],c^co' 
{k, -mco^ + r,) + i{c,(0-rij) 


tC", 


(30b) 


where Eqns. (15) and (16) have been applied in all Cases. Note that in Eqn. (27a), it is assumed 
that the rotational spring at the support is attached to the cross section of a shaft element such 
that the rotational spring responds only to the slope change due to rotation of the cross section 
and not the total slope change of the neutral axis of the shaft model. This assumption allows the 
shearing motion of the shaft element at the support. Note also that the effect of axial loads on the 
shear force at the support is neglected since the contribution of axial loads to the shear force at 
the support or boundary is small compared to the shear force due to the flexural motion of the 
shaft element. Exact moment and force balance conditions at boundaries for a rotating 
Timoshenko shaft element subjected to axial loads can be found in Ref. [16]. 

The corresponding matrix equations for the simple Euler-Bernoulli shaft model are shown in 
Appendix I. Solving the set of matrix equations simultaneously for r and t gives the elements of 
the reflection and transmission matrices for each Case. The general forms of solutions to these 
sets of equations for each Case is not presented in this paper due to space limitation. However 
one can obtain the solutions in either closed-form or numerically. Note that in Case II and Case 
IV, the first columns of r and t are the reflection and transmission coefficients due to incident 
propagating wave components, and the second columns are due to an incident attenuating wave 
component which is generally termed as near-field since this type of wave decays exponentially 
with distance. When the distance between the origin of disturbance and the discontinuity is very 


1301 



large, these attenuating wave components can be neglected. However, as mentioned by many 
authors, for example Graff [2], attenuating waves play an important role in wave motions by 
contributing a significant amount of energy to the propagating wave components when a set of 
propagating and attenuating waves are incident at a discontinuity and, in particular, when the 
distances between the discontinuities are relatively small, as in the case of closely-spaced multi¬ 
span beams. In this paper, near-field components are included. In what follows, the effects of 
the point supports on the reflection and transmission of an incident wave are studied. For 
comparison, the results are obtained for both the Timoshenko and the simple Euler-Bemoulli 
models, which hereafter, for brevity, are denoted by TM and EB, respectively. The system 
parameters used in the numerical results are taken from Ref. [10]; ao - 0.0955 m, p = 7700 
kgW, K = 0.9, E = 207x10^ Wnf, G = 77.7x10^ N/ml 


3.1. Wave reflection and transmission at rigid supports 

Consider two cases: the simple support and the clamped support. The r and t are solved and 
shown as follows. 

• Simple support (k, = oo, = m= c, = - 7„, =0) 

Case 7 (A > 0 and B > 0): 


r = 


_ 1 _ 

(72-7i)(r 1 / 2 + 6 ;') 


7i(6>"-72) 
72 ( 7 ?-6;-) 


7i(6;’-72) 


(31a) 


t = 


_ 1 _ 

(72-7i)(7i72+6)-) 


72 ( 7 ?-®') 


Case 77 (A > 0 and S < 0): 

_ 1 _ 


_ 1 _ 

(iT,-r,xr,r,-!V) 




Case 7V(A<0andB<0): 


_ 1 _ 


7i(6)"“72) 
7i(72-6)') 


r,(r,H®-) 1 


r,{r^ + co^)' 


(31b) 


(32a) 


(32b) 


(33a) 


1302 




The corresponding reflection and transmission matrices for the EB model are listed in App. I 

Figures 3 and 4 plot the moduli (magnitudes) of the reflection and transmission coefficients 
for the simple and clamped supports. The finite cutoff frequencies, above which all waves 
propagate, are also marked in the figures. Thus, for the TM model, the wave motions change 
from Case 11 to Case 1 when (0>C0^ {0)^-4- in Fig. 3, co, = 4.24 in Fig. 4; O), is slightly altered 
by rotation speed and axial load). The results show that, at low frequencies (cD < 0.1 = 3156 
rad/sec), the wave reflection and transmission coefficients of the TM model agree well with those 
of the EB model for both support conditions. However, as the frequency increases, the wave 
propagation characteristics of the TM model differ significantly from those of the EB model. 
These differences can be explained by examining the different modes of vibration. When 
co>co^ (in the regime of Case 1), the vibrating motion of the TM model is dominated by the pure 
shearing motion [7, 17], and hence the EB model, which neglects the rotary inertia and pure 
shear effects, become inaccurate at high frequency. As discussed in Ref. [7], at the finite cutoff 
frequency, the TM shaft experiences no transverse displacement, and the cross-section of the 
shaft simply rotates back and forth in unison 

In Figs. 3(d)-(f) and 4(d)-(f), for = 0 and £ = 0, the reflection and transmission coefficients 
of the EB model are independent of the frequency. This is because from Eqn. (10a), A = 0, and 
Eqns. (9c,d) lead to a single wavenumber F, = Tj. From Appendix I, Eqns. (32*-36*), the r and 
t are thus constant matrices. It is also seen that the wave reflection and transmission coefficients 


1303 




for both shaft models are basically independent of the rotation speed over the entire frequency 
range, even at high rotation speed = 0.05 = 44,600 rpm. In Ref. [7], it is also found that has 
negligible effects on the system frequency spectrum, phase velocity and group velocity. On the 
other hand, the effects of the axial load are significant for both propagating and attenuating 
waves in the regime of Case H, see Figs. 3(b)-(c) and 4(b)-(c). For both shaft models under 
simple support and compressive loads (Figs. 3(b, e)), the reflection coefficient ru of the incident 
propagating wave is reduced significantly in the regime of Case II, while the transmission 
coefficient fj, of the propagating wave component increases to balance the energy carried in the 
wave. However, the attenuating wave component which does not carry any energy loses its 
transmissibility in the same amount as the reflection coefficient r, 2 . Thus, in the presence of a 
compressive load, most of the transmitted wave energy in Case II comes from the propagating 
component of the incident wave. Note that axial tensile loads have the reverse effects on these 
wave components. In the clamped support case, the positive propagating wave component rn is 
constant over the regime of Case II under any loading conditions for the both shaft models, as 
seen in Fig. 4. 

Since there is no damping at the support, the incident power (Ilinc), reflected power (Hrefi) and 
transmitted power (Iltnin) in Cases II and IV are related by flinc = rirefl+ritran = Hinc, 

or Iriil^+Uiil^ = 1. This relationship is confirmed by the plots shown in Figs. 3 and 4, where for 
both shaft models, Irni and knl cannot exceed one. However in the regime of Case I, in which 
all wave components propagate, the energy balance is Hinc = (l^ii+r 2 iP+lfii+r 2 !p) Ilinc, or Hinc = 
(Iri 2 +r 22 l^+l?i 2 +r 22 l^) Hinc- Together with the plots on the phase of these coefficients (not shown 
to minimize the size of this manuscript), the above relationships can also be verified for wave 
motion of Case I. 


1304 












support without “resistance”. The impedance mismatching (rn = 1, ?,, = 0) frequency at which 
the propagating wave component is completely reflected without being transmitted can also be 
determined from Figs. 4(b, d) for the two shaft models. This impedance mismatching frequency 
is located in the regime of Case II for the TM model where the transverse mode dominates the 
vibrating motion of the shaft. Numerical results show that, as the spring constant increases, this 
impedance mismatching frequency increases, but is limited to within the regime of Case II and 
can never be found in the regime of Case I where the pure shearing mode dominates the vibrating 
motion of the shaft (refer to Fig. 3 for the transition of types of wave motion). 

Figure 6 shows the reflection and transmission coefficients for waves incident upon a support 
having both translational and rotational constraints. Since both flexural and shearing modes of 
vibration are constrained at this support, the maximum of the reflection coefficient is expected to 
be higher than the previous case. Figures 6(a-b) and (c-d) are the results for the TM and EB 
models, respectively. The translational and rotational spring constants used in the simulations 
are k,Q- 10^ N/m and ^^ 5 = 10^ Nm/rad, respectively. It is noted that in the regime of Case II, i.e.. 



Figure 6. Wave reflection and transmission coefficients at an elastic support with translational and rotational springs 
{k,-k,Q, k^^O, c, = c, = m= y,„ = 0) as a function of frequency, J3 = 0.05 and e = 0. (a-b) and (c-d) are results 
for the Timoshenko and Euler-Bernoulli shaft models, respectively. 





modulus modulus 


2.0 


r,2 (upper line) 

r,2=:r2, (middle dashed line) 

Tj, (lower line) 



Figure 4. Wave reflection coefficients at a clamped support (/c, = fc, = ■» and = c, = m = = 0) as a function of 

frequency, (a)-(c) and (d)-(f) are the results for the Timoshenko and Euler-BernouUi shaft models, respectively. The 
transition from one type of ivave motion to another is marked for the case /J = 0.05, £ = -0.05. 





3.2, Wave reflection and transmission at elastic supports 

Figure 5 shows the reflection and transmission coefficients for waves incident upon a support 
with a finite translational spring for three different spring constants. Figures 5(a)-(b) and (c)-(d) 
are results for the TM and EB models, respectively. The spring constant used, k^Q = 10^ N/m, is 
a typical bearing spring constant value for turbine generators. The plots show that there is no 
significant difference in the moduli between the two shaft models. This is because the incident 
wave does not experience any rotational constraint at the support, and hence the additional rotary 
inertia factor in the TM model has only a small contribution to the wave motions. As the support 
spring constant increases, the curves for both the reflection and transmission coefficients are 
shifted to the right and, as the spring constant approaches infinity, these curves eventually 
become asymptotic to those shown in Fig. 3. Note that an impedance matching (r = 0, t = I), 
where all wave components are transmitted without being reflected, is found in the high 
frequency region for both shaft models. Thus, as the frequency increases, the characteristics of 
waves travelling along the shaft remain unchanged such that waves propagate through this elastic 



Figure 5. Wave reflection and transmission coefficients at an elastic support with a translational spring 

= c, = c, =m = =0) as a function of frequency, /3 = 0.05 and £ = 0. (a-b) and (c-d) are results for the 

Timoshenko and Euler-Bernoulli shaft models, respectively. 


1308 





in the low frequency range, both shaft models have similar reflection characteristics, and both the 
reflection and transmission coefficients are not significantly affected by the rotational spring. 
However, as the frequency increases, the effect of the rotational constraint on the wave motion 
becomes eminent, particularly for the TM model. As seen in Figs. 6(a) and 6(c), the reflection of 
the attenuating wave components are significantly higher than those of the propagating wave 
components. Hence, when a rotating shaft has a clamped support(s) such as a journal bearing, 
contributions from the attenuating wave components should be included in the formulation since 
a significant amount of energy in the propagating component arises from the incident attenuating 
wave component. It is noted that the impedance matching regions seen in Figs 5(a, c) disappear 
when the rotational constraint is added. Moreover, the impedance mismatching frequency shown 
in Figs. 5(b, d), which is found in the regime of Case II, also does not occur. At low frequency in 
Figs. 6 (b, d), there appears to be a mismatching region, but rj, is not exactly equal to zero. 
From Figs. 6(a, c), it is seen that there is a frequency at which the positive propagating wave 
component r,, is zero (this frequency is slightly different for the two models). This frequency 
does not correspond to an impedance matching, though the propagating wave is not reflected at 
all but is only transmitted (r,i = 1). Based on other research results [18], this phenomenon likely 
indicates a structural mode delocalization in bi-coupled systems, in which vibrations on both 
sides of the support become strongly coupled. Further research on the vibrations of rotating 
shafts with intermediate supports is being pursued to confirm the mode delocalization. 

Figure 7 plots the effects of axial compressive loads on the wave reflection and transmission 
upon a support with finite spring constant for the Timoshenko shaft model. As seen in Fig. 7(a), 
the reflection coefficient for the incident propagating wave component ru is substantially 
reduced in the low frequency range while the reflection coefficient for the incident attenuating 
wave component increases significantly. However, Fig. 7(b) shows the reversed effects on the 
transmission coefficient. It can therefore be concluded that, when the shaft is axially strained by 



Figure 7. Wave reflection and transmission coefficients at an elastic support (A:^ = k,Q and kr = c, = Cr = m = J„, = 0) 
for the Timoshenko shaft model with and without the compressive load, (a) reflection coefficients, (b) transmission 
coefficients. 


1309 




Figure 8. Wave reflection and transmission coefficients at an elastic support k^ = and c,= Cr = m = 

J„, = 0) for the Timoshenko shaft model with and without the compressive load, (a) reflection coefficients, (b) 
transmission coefficients. 


compressive loads, the energy contribution from the incident attenuating wave component to the 
energy in the reflected propagating wave is more significant than the strain-free situation in the 
low frequency range, while most of the energy in the transmitted wave derives from the incident 
propagating wave component. 

Figure 8 plots the wave reflection and transmission coefficients along an axially compressed 
Timoshenko shaft model at a support with finite translational and rotational spring constants. 
Similar results to the previous example can be observed in terms of energy contribution from the 
incident attenuating wave component in the low frequency range. However, the effects of the 
axial compressive load on both the reflection and transmission coefficients for the propagating 
wave component ( r^^ and r,,) are significantly reduced when compared to Fig. 7. 


3.3. Wave reflection and transmission at damped supports 

Figure 9 shows the effects of both translational and rotational dampers at a support with finite 
translational and rotational spring constants. Figures 9(a, b) and (c, d) are results for the TM and 
EB models, respectively. The translational and rotational damping constants used in this study 
are c^to = 2x10^ Ns/m and Cdro = 64x10^ N-m-s/rad, typical values for bearings in turbine 
generators. The curves with symbols (• and ♦) are the results when the rotational damping factor 
is also included in the formulation. It can be seen that I and Ir^l for both shaft models are 
significantly lowered due to the presence of damping. Note that, because of the damping, the 
frequency at which 1 rj, I = 0 (compare with Figs. 6(a, c)) no longer exists for both shaft models. It 
can also be seen that the effect of the rotational damping factor on the wave reflection and 
transmission is not significant over the entire frequency range for both shaft models. For TM 
model, the contribution of the rotational damping to both Ir;, 1 and 1?,, I is almost negligible. The 


1310 


support condition considered in this particular example is simulated as an actual bearing support 
adopted in turbine generators. Hence for this particular type of bearing support, the effect of the 
rotational damping on wave reflection and transmission is not considerable. Other numerical 
results (not shown in this paper) show that the wave propagation at the damped support is 
characterized by translational damping rather than rotational damping. Note that similar results 
have been presented for the support without damping (see Fig. 6). 



Figure 9. Wave reflection and transmission coefficients at an elastic support with damping (k, = k,o, K = Ko, c, = c,;,o, 
c, = cjro and m = J,„ = 0) as a function of frequency for /3 = 0.05 and £ = 0. (a-b) and (c-d) are results for the 
Timoshenko and Euler-Bernoulli shaft models, respectively. 


3.4. Wave reflection and transmission at a rotor mass 

Consider a gear rigidly assembled to a rotating shaft. The gear is assumed to be perfectly 
balanced and its thickness is sufficiently small such that wave reflection and transmission due to 
the geometric discontinuity between the shaft and the gear can be neglected. However the gear 
does resist the translational and rotational motions of the cross-sectional element of the shaft. 


1311 





Figure 10 shows the reflection and transmission upon the gear when the mass mo and mass 
moment of inertia J^o of the gear are 4 and 16 times of the shaft, respectively. Not shown in 
Figs. 10 (b, d) is that r ,2 = 0 when w= /„ = 0. Like some previous support conditions 

discussed, the effects of the rotor mass are much more significant in the high frequency region 
for both models (particularly around and beyond the cutoff frequency for the TM model). In 
general, the rotor mass decreases the transmission and increases the reflection of the wave. At 
very high frequency, there is basically no wave transmission. Note that, since the geometric 
discontinuity between the shaft and the gear is neglected in this model, one may expect that the 
actual reflection for both the propagating and attenuating wave components would be higher. 


(a) 


(m= = V 




(b) 


-(m = 0, 

-( m = mg. ) 


/ 




Figure 10. Wave reflection and transmission at a rotor mass assembled to a rotating shaft (*, = 0 = L = Cf = Cr = 0, 
and m = rriQ, and J,„ = J„^) as a function of frequency when P = 0.05 and £ = 0. (a-b) and (c-d) are results for the 
Timoshenko and Euler-Bernoulli shaft models, respectively. 


4. WAVE REFLECTION AND TRANSMISSION AT A GEOMETRIC DISCONTINUITY 
It is common for a rotating shaft element to have changes in cross-section, or to be joined to 


1312 






Case I, II, or IV 

A, = (1 + a)co- - 2j3co - 16£ (1 + £ - ~) 

a 


Z=0 


Case I, II, or IV 


B, = co' 


ccco- - 2p(o - I6a (1 + £)(1 + £-) 

a 


A={\ + a)(o^-2^co-^^^ (l + £, -—) 
o" a 

16a 


= co^\a (O' -215(0-^^ (l+£j(l + £ -— ) 

L O’ a 


Figure 11. Wave reflection and transmission at a geometric discontinuity. 


another shaft element by a coupling. Figure 11 shows a typical example of a discontinuous shaft 
model in which two shafts of differing wavenumber and diameter are joined at z = 0. The 
subscripts I and r denote z = O' and z = 0^ regions, respectively. It is known that when a wave 
encounters a junction or a discontinuity, its wavenumber is changed. It is therefore possible that 
a wave on the left side of the junction can be propagating, while after crossing the junction to the 
right side, the wave becomes attenuating. Therefore, for a Timoshenko shaft, when a wave 
propagates through the junction, there are mathematically nine possible different combinations of 
wave motions to be considered depending on the values of the functions A and B on each side of 
the junction, as depicted in Fig. 12. 



Figure 12. Nine possible combinations of wave motions at a geometric discontinuity of the cross section for the 
Timoshenko shaft model. Subscripts / and r denote the left and the right side of the discontinuity, respectively. 


1313 




For simplicity, assume that material properties such as p, E, and G are the same for both sides 
of shaft element. The displacement continuity, moment and force equilibrium conditions are 
applied at the junction to determine the wave refection and transmission matrices. Results for the 
three most commonly encountered possibilities in the low frequency regime are listed as follows. 

Case II {A, >0, Bi <0) - Case I (A^ > 0, R, >0): 



■ 1 r 



1 

1 ' 


r 1 M 





rC" = 


tCY 


Jlu ^21. 



nu ■ 

~n2i. 


Jl\r ^2r_ 


r -i^Bu 

^21^2! 

C" + 

r Xtnu 

”Y,r72, 1 

L^xr., 

-riu) r 

21-in 21^ 

1 

Y 

rr 

1 

T 

^772,)J 


rC-^ 


Yuriu 72rn2r 

L-io-"(r„-77„) -i<y^(r2r-V2r)J 


tc", 


Case II (A, >0, B, <0) - Case II (A^ > 0, B^<0): 


■ 1 1 ■ 


■ 1 1 ■ 


■ 1 1 * 

C" + 

rC^ = 

Jlv n2l. 


— nu “^2/. 


Y.r n2r. 


(37a) 


(37b) 


(38a) 




-"^21^21 

^2, -in21 


C" + 


i^unu ^21^21 

~i(^H — nu ) ~(^2/ ~ ^^21 ) 


rC" 


-i<yXrnu -^Xr'n2r 
-zcr"(r,,-77„) -C7“(r2,-7]2,)J 


CaseII{A^>0, B, <0) - Case IV {A, <0, S, <0): 


r i 1 1 


r 1 11 


■ 1 r 


C" + 


rC^ = 


nu n 2 i 


l-nu - n 2 i ^ 


n 2 r n ^ r . 


Y/^2/ 



^2/^2/ 

JXi—nu) Y/~^*^2;. 


“(Yf -in2i)_ 


-i^%rn2r 


-<yXrn:r 


tc\ 


(38b) 


(39a) 


(39b) 


where cr is the diameter ratio between the shaft elements, defined as 


1314 


( 40 ) 


CJ = —. 


Note that 77/5 in Eqns. (37a-39b) are given by Eqns. (20a, b), (22a, b), and (24a, b) according to 
the type of wave motion, and 77 / s on the right side of the geometric discontinuity are modified 
as follows. 


where 


: - 77 =-— for Case I, 

r„e; yzX 

(41a, b) 

-co^ Tl+co- 

, Ti, - — - 7 - for Case II, 

r„e; iT^rS; 

(42a, b) 

,7]2 for Case IV, 

ir,x r,,e; 

(43a, b) 

£' = 1 + 8,. —and £, = — . 

(44) 


a O' 


Moreover, the wavenumbers, A and B of the shaft element on the right side of the junction are 
modified as follows. 

y„ = ^[a, + ^Aj-4B^f . r,, = - Va?- 4B,)^ , (45a. b) 

+ = . (45c,d) 

where, 

A. = (l + a)a.= -2pa)-^a + £,-|), 


B. =co‘ 


am--2Pa-^ (l + £,)(l + £,-^) 

C OL 


(46b) 


Corresponding results for the simple Euler-Bemoulli shaft model are listed in Appendix II. 

Figures 13 to 16 show some representative examples of wave reflection and transmission 
upon the geometric discontinuity. In Figs. 13 and 14, the thick and thin curves represent results 


1315 


for the TM and EB models, respectively. The second graph in each figure shows the changes of 
Ai, Bi, Ar and Br, and how wave solutions on both sides of the discontinuity change as the 
frequency increases for the TM model. In general, the wave reflection and transmission for the 
EB model are frequency independent except when the shaft is axially strained, while the wave 
propagation characteristics for the TM model are strongly dependent on the frequency. 

Comparing Figs. 13 and 14, it is noted that, for both shaft models, the average reflection and 
transmission rates for cr = 0.8 are higher than those for o' = 1.2, especially for the attenuating 
wave components. These results imply that incident attenuating waves contribute more energy to 
propagating waves at the discontinuity when the waves travel from a smaller to a larger cross- 
section. In particular, it is noted that the transmissibiiity of the attenuating wave tn has a strong 
dependency on the direction of propagation. Note also that the differences between the two shaft 
models are more pronounced when cr = 0.8. It is clearly seen from the figures that when Bj and 
B^ change from negative to positive, both reflection and transmission coefficients experience a 
sharp jump or drop at the finite cutoff frequencies, due to changes in the types of wave motion. 
In the frequency region (S, > 0 and B^ <0) located between the two cutoff frequencies in Fig. 


2.0 h 


0.5 


Thick curves : Timoshenko shaft model 
Thin curves : Euier-Bernoulli shaft model 


- Ui 




h 

i 

il _ 


- 

7/ X.-V' 




.'7 


\ 


5 

(O 



Figure 13. Reflection and transmission of waves incident upon a change in the cross-section, a= 0.8, P = 0.05, and 
£ = 0. Thick and thin curves are results for the Timoshenko and Euier-Bernoulli shaft models, respectively. Note 
that the ordinates in the lower graphs keep increasing with frequency (abscissa). 


1316 






13, the wave motion on the left side of the junction is governed by the wave solution of Case I 
since all wave components are propagating at a frequency larger than the cutoff frequency, while 
the wave motion on the right side of the junction is governed by the wave solution of Case II. 
Thus, for O’ = 0.8, some of the propagating wave components on the left side of the shaft element 
cannot propagate as they pass the discontinuity, and become attenuating. A similar, but converse 
conclusion can be drawn for the frequency region (5^ > 0, 5, < 0) when cr = 1.2, as shown in 
Fig. 14. The results of Figs. 13 and 14 show that, for different system parameters cr, jS, and e and 
at any given frequency, the types of wave motion on each side of the discontinuity can be 
different, as depicted in Fig. 12. 

From Eqns. (41a-43b), it is seen that when the Timoshenko shaft is axially strained and (O is 
not sufficiently large, the wavenumber (hence wave propagation characteristics) depends strongly 
on the cross-section ratio <7. Figure 15 shows the effects of the axial load on the wave reflection 
and transmission, which are mostly limited to the relatively low frequency region. In Figs. 15(a- 
b), when the shaft is axially compressed (£ = -0.05), the reflection and transmission due to the 
incident attenuating wave component decrease for both o < 1 (plot (a)) and <7 > 1 (plot (b)). 
However, the transmission due to an incident propagating wave decreases significantly for 



— — f, 2 


2.0 h 



Figure 14. Reflection and transmission of waves incident upon a change in the cross-section, cr= 1.2, /3 = 0.05, and 
£ = 0. Thick and thin curves are results for the Timoshenko and Euler-Bernoulli shaft models, respectively. Note 
that the ordinates in the lower graphs keep increasing with frequency (abscissa). 


1317 









Figure 15. Reflection and transmission of waves upon a change in the cross-section when = 0.05 for the 
Timoshenko shaft model, (a) < 7 = 0.8 and £ = -0.05. (b) (T= 1.2 and e=-0.05. (c) cr= 0.8 and £= 0.05. (d) or = 1.2 
and e= 0.05. Thin and thick curves show the results when the shaft is strain-free (£= 0) and strained, respectively. 


c - 0.8 and increases for <7 = 1.2 at low frequency. 

Effects of the axial load on the wave reflection and transmission are more significant when 
the shaft is compressed (Figs. 15(a-b)) than when it is under tension (Figs. 15(c-d)). This is 
because the wavenumbers of both the propagating and attenuating wave components are only 
slightly changed. It is also noted that, in the low frequency range, the wave solution of Case IV 
governs the wave motions on both sides of the discontinuity, and the wave components which 
have large wavenumber (Fi) attenuate, while wave components with small wavenumber (r 2 ) 
propagate along the waveguide as long as A remains negative. 


1318 








5. WAVE REFLECTION AT BOUNDARIES 


When a wave is incident upon a boundary, it is only reflected because no waveguide exists 
beyond the boundary. Consider an arbitrary boundary condition with translational and rotational 
spring constraints, dampers, and a rotor mass, as shown in Fig. 16. The reflection matrix at the 
boundary is derived for each Case. Applying the same non-dimensional parameters employed in 
Section 3, and by imposing the force and moment balances at the boundary, which can be 
deduced by eliminating and inEqns. (27a, b), 

M~ = k^y/ + c^yr + J„y/, (47 a) 

-V = k,u + c,u + mu, (47b) 

the reflection matrix for each Case is determined. 

CaseI{A>0, B>0): 

J r72(ir2-2:,J T’r-T7,(jri + ^.) + 

+ Kr2-Bi) + ^s\ Kr2-T72)-^.J’ 


Case 7/ (A > 0, 5 < 0): 

7],(ir,-2:j T'r-77,(ir,+Ej -7]2(r2 + zj1 

'''[i(r,-r7,)+z, (r,-ii7,)+i.J [i(r,-j?,)-z, 


M,Jm 



Z=0 


Figure 16. Wave reflection upon a general boundary. 


1319 


CaseIV{A<0, 5<0): 


772 (^r2 s„,) 

iCFj - 7 ) 2 ) + 


(r.-mj+s. 


772 (zTj + Sp,) 

/(r2 ~ ^ 2 ) ■“ 


-^iCr.+zj 

(r,-j77,)-E, 


(50) 


where 77 's in above equations have been defined in Eqns. (20a, b), (22a, b) and (24a, b), and 

Z„, = + ic^co - J,„co ~, and = ^, + ic,Q) - mco^ , (51a, b) 

The corresponding results for the simple Euler-Bernoulli shaft model are listed in the Appendix 
in. By specifying the parameters in the reflection matrix r, results for three typical boundary 
conditions (simple support, clamped support, and free end) can be obtained. 

• Simple support {k, =^, =m = c, = = 7,„ = 0) 


r 


-1 0 
0 -1 


for Case I, II, and IV, 


(52) 


• Clamped support (k^ = k,. = m = c, = = J„, = 0) 

1 

Tli-ri2 


(53b) 

(53c) 

• Free end {k, = k^ = m = c, = = /„, = 0, and £ = 0) 


1 


rii+ri2 


2772 


77,- 772 L-277i -(r?,+772)J 

1 




-irii+Tli) "2772 

2t}, ( 771 + 772 ) 


for Case II, 


for Case IV, 


■ 77 ,+7?2 2772 

- 277 , -( 771 + 772 ), 


for Case I, 


(53a) 


r 


J_ 


77,7?2(ri+72)”7ir2(77,+ 772 ) 
-277,7i( 77,-7i) 


277272 ( 772 - 72 ) 

- 77 , 772 ( 71 +72) + 7i72(77i+ 772 ). 


(54a) 


where. A, = r],n, - 7, ) + 7,72 (77, - ^2 ) for Case I, 


+ 72 ) + 7 i 72 ( 77 i + 772 ) 2i7}^y^iin^-y^) 

^ A;, [ 2/77,7,(771-7,) 77,772(/7,+72)-7 i 72(7?,+772) 


(54b) 


where, = 77 , 772 ( 77 , - 72 )- 7 , 72 ( 7 ?, - 772 ) for Case//, 


1320 



Figure 17. An example of a rotating shaft with multiple supports and discontinuities. 


I \TiiV 2 (ri+ir 2 )-rj 2 (Tii+r] 2 ) 277,7,(77,+ir,) ' ^ 

r =- (54c) 

A/v L -2i7?2r,(77j-7j - 77 , 772 ( 7 , +J72)+rir2(^i +^ 2 ). 

where, A;^ = 77 , 772(71 - 172 ) “7172(Hi- 772 ) for Case IV. 

6. APPLICATIONS 

The reflection and transmission matrices for waves incident upon a general point support or a 
change in cross-section can be combined with the transfer matrix method to analyze the free 
vibration of a rotating Timoshenko shaft with multiple supports and discontinuities, and general 
boundary conditions. The basic idea of this technique has been shown in Ref. [6]. However, due 
to the complex wave motions in the Timoshenko shaft model, such as the frequency dependency 
of the wave reflection and transmission at a cross-section change, it is important to apply the 
proper reflection and transmission matrices consistent with the values of A and B on both sides of 
the discontinuity, particularly when numerical calculations are performed. Consider for example 
the free vibration problem of the rotating Timoshenko shaft model shown in Fig. 17. Denoting R 
as a reflection matrix which relates the amplitudes of negative and positive travelling waves at a 
discontinuity, and defining T/ as the field transfer matrix which relates the wave amplitudes by 

C^iZo + z) = TC"(Zo), C-(zo + z) = T-'C-(zo), (55) 

the following relations can be found. 


1321 



W-=R5W^ 

(R5=>-s). 

(56a) 


i = 2,3,4 (station number), 

/i = left (/) or right (r) 

(56b) 

w>T,w-,, 


(56c) 

< =r,w-, 


(56d) 

<=T.w;, 


(56e) 


where in Eqn. (56b), 

R,,=TiR„,,T,. R,,=r,+t,(R-’-r,-)-'t, (56f) 


Solving the above matrix equations gives 

(r,T,R„T,-I)w>0, (57) 

where each element of the matrix is a function of two different wavenumbers and the frequency 
CO. For non-trivial solutions, the natural frequencies are obtained from the characteristic equation 

Det[(r,T,R„T,-I)] = 0. (58) 

The proposed method is applied to an example of a two-span rotating shaft, simply supported 
at the ends and with an intermediate support consisting of translational and rotational springs, as 
shown in Fig. 18. Numerical computations were performed by a PC-based Mathematical. The 
values of the spring constants Kj and Kr are those introduced in Section 3.2, with ^ = 1 m and 
the rotation parameter p = 0.05. 



1322 


Figure 19 shows the first eight natural frequencies of the vibrating shaft for both the TM and 
EB models for a classical simple intermediate fixed support K^=0) placed at 

various locations. The results confirm the well-known fact that the Timoshenko model leads to 
smaller eigenvalues. Figure 20 shows the first eight natural frequencies of the Timoshenko shaft 
for an elastic intermediate support with three different translational spring constants. It can be 
seen that the effect of the translational spring diminishes for higher modes. The proposed wave 
analysis technique can also be applied effectively to the study of structural mode localizations in 
mistuned, rotating systems. Dynamics of such systems will be addressed in another paper. 




Figure 19. Natural frequencies of a two-span, rotating 
shaft as a function of the support location; intermediate 
support is fixed. 


Figure 20. Natural frequencies of a two-span rotating 
Timoshenko shaft as a function of the support location; 
intermediate support consists of k, and kr. 


7. SUMMARY AND CONCLUSIONS 

In modern high speed rotating shaft applications, it is common that the shaft has multiple 
intermediate supports and discontinuities such as bearings, rotor masses, and changes in cross- 


1323 










sections. In many cases, the ratio of the shaft diameter to its length between consecutive supports 
is large, and the Timoshenko model (TM) is needed to accurately account for the shear and rotary 
inertia effects. In this paper, the wave propagation in a rotating, axially strained Timoshenko 
shaft model with multiple discontinuities is examined. The effect of the static axial deformation 
due to an axial load is also included in the model. Based on results from Ref. [7], there are four 
possible types of wave motions {Cases I, II, III and IV) in the Timoshenko shaft, as shown by 
Eqns. (5a-8b). In practice. Case III does not occur and is excluded in the analysis. For each 
Case, the wave reflection and transmission matrices are derived for a shaft under various support 
and boundary conditions. Results are compared with those obtained by using the simple Euler- 
Bemoulli model (EB) and are summarized as follows. 

1) In general, the two shaft models show good agreement in the low frequency range where the 
wave motion is governed by Case II and Case IV. However, at high frequencies, the types of 
wave motions and propagation characteristics for the TM and EB models are very different. 

2) The effects of shaft rotation on the wave reflection and transmission are negligible over the 
entire frequency range and even at high speed (up to 44,600 rpm). While the effects of the 
axial load are significant, especially in the low frequency range. 

3) When waves are incident at supports with only translational springs, differences in the results 
between the TM and EB models are small, and there exists frequency regions of impedance 
matching and an impedance mismatching frequency (limited to within the regime of Case IT). 
The impedance matching and mismatching disappear when a rotational spring is added to the 
support. Instead, there is a frequency at which Irul = 0 and Ifni = 1, and vibrations on both 
sides of the support become strongly coupled. This (delocalization) phenomenon suggests 
further research on the vibrations of constrained multi-span beams. When there is damping 
at the support, the frequency at which IrnI = 0 does not occur. Moreover, effects of 
translational damping on the wave propagation are more significant at high frequency, 
especially for the TM model, however effects of rotational damping is not significant over the 
entire frequency range. 

4) Contributions of attenuating wave components to the energy in the reflected and transmitted 
waves are significant when the shaft is axially strained and when the support has a rotational 
constraint. Thus attenuating waves should be included in the formulation. 

5) Unlike the spring supports, in which waves are easily transmitted at high frequency, the rotor 
mass support diminishes the wave transmission as the frequency increases. 

6) When waves are incident at a geometric discontinuity such as a change in the cross-section, 
there are nine possible combinations of wave motions on both sides of the discontinuity. It is 
shown that differences of the results between the TM and EB models depend on the diameter 
ratio (and hence the direction of the wave incidence). Moreover, incident attenuating waves 
contribute more energy to propagating waves at the discontinuity when the waves travel from 
a smaller to a larger cross-section. When the shaft is axially strained, the effects of the load 
on the wave propagation are primarily limited to the low frequency range. 

The reflection and transmission matrices are combined with the transfer matrix method to 
provide a systematic solution method to analyze the free vibration of a multi-span, rotating shaft. 
Since the procedure involves only 2x2 matrices (while including the near-field effects already), 
strenuous computations associated with large-order matrices are eliminated. 


1324 


ACKNOWLEDGMENTS 


The authors wish to acknowledge the support of the National Science Foundation and the 
Institute of Manufacturing Research of Wayne State University for this research work. 


REFERENCES 

1. Lin, Y.K., Free Vibrations of a Continuous Beam on Elastic Supports. International Journal 
of Mechanical Sciences, 1962, 4, pp. 409-423. 

2. Graff, K.F., Wave Motion in Elastic Solids, Ohio State University Press, 1975. 

3. Cremer, L,, Heckl, M. and Ungar E.E., Structure-Bome Sound, Springer-Verlag, Berlin, 
1973. 

4. Fahy, F., Sound and Structural Vibration, Academic Press, 1985. 

5. Mead, D.J., Waves and Modes in Finite Beams: Application of the Phase-Closure Principle. 
Journal of Sound and Vibration, 1994, 171 , pp. 695-702. 

6. Mace, B.R., Wave Reflection and Transmission in Beams. Journal of Sound and Vibration, 
1984, 97 , pp. 237-246. 

7. Kang, B. and Tan, C.A., Elastic Wave Motions in an Axially Strained, Infinitely Long 
Rotating Timoshenko Shaft. Journal of Sound and Vibration (submitted), 1997. 

8. Han, R.P.S. and Zu, J.W.-Z., Modal Analysis of Rotating Shafts: A Body-Fixed Axis 
Formulation Approach. Journal of Sound and Vibration, 1992,156, pp. 1-16. 

9. Zu, J.W.-Z. and Han, R.P.S., Natural Frequencies and Normal Modes of a Spinning 
Timoshenko Beam With General Boundary Conditions. Transactions of the American 
Society of Mechanical Engineers, Journal of Applied Mechanics, 1992, 59, pp. 197-204. 

10. Katz, R., Lee, C.W., Ulsoy, A.G. and Scott, R.A., The Dynamic Response of a Rotating 
Shaft Subject to a Moving Load. Journal of Sound and Vibration, 1988,122, pp. 131-148. 

11. Tan, C.A. and Kuang, W., Vibration of a Rotating Discontinuous Shaft by the Distributed 
Transfer Function Method. Journal of Sound and Vibration, 1995,183, pp. 451-474. 

12. Argento, A. and Scott, R.A., Elastic Wave Propagation in a Timoshenko Beam Spinning 
about Its Longitudinal Axis. Wave Motion, 1995, 21 , pp. 67-74. 

13. Dimentberg, F.M., Flexural Vibrations of Rotating Shafts, Butterworth, London, 1961. 

14. Dimarogonas, A.D. and Paipeties, S.A., Analytical Method in Rotor Dynamics, Applied 
Science, New York, 1983. 

15. Lee, C.W., Vibration Analysis of Rotors, Kluwer Academic Publishers, 1993. 

16. Choi, S.H., Pierre, C. and Ulsoy, A.G., Consistent Modeling of Rotating Timoshenko Shafts 
Subject to Axial Loads. Journal of Vibration and Acoustics, 1992, 114 , pp. 249-259. 

17. Bhashyam, G.R. and Prathap, G., The Second Frequency Spectrum of Timoshenko Beams. 


1325 


Journal of Sound and Vibration, 1981,76, pp. 407-420. 

18. Riedel, C.H. and Tan, C. A., Mode Localization and Delocalization of Constrained Strings 
and Beams. Proceedings ofASME Biennial Conference on Mechanical Vibration and Noise 
(submitted), 1997 



1326 


For simple and clamped supports, the reflection and transmission matrices are listed as follows. 


Simple support (k. = k. = n 

l = C, 

= c, = . 




Case // (A > 0, B < 0): 

1 

■ r. 

r, ■ 

iF, r2 

'ir, 

r./ 

'"ir.-r2 



.-ir^ 

-r^. 

CaseIV{A<0, B<0): 

1 


-ire 

1 1 

t “ 

'r. 

-ir: 

iT, +r2 

.-^2 

-n_ 

i ^T,+r2 

-r2 



• Clamped support {k^ =<=<>, m=c, = c, = J„, = 0); t = 0. 

CaseII{A>0, B<0): 

1 ^^ 1+^2 2 r 2 

-2iT, -(ir, + r,)J’ 

CaseIV{A<0, B<0): 

1 r-(ir,-r2) -2iTi ' 

^“iT.+r.L -2r2 iT,-r^ ■ 


(32a*, b*) 


(33a*, b*) 


(35*) 


(36*) 


APPENDIX II 

The reflection and transmission matrices for a wave incident upon a cross-sectional change 
for the simple Euler-Bernoulli shaft model can be determined by solving the following sets of 
matrix equations. Only two representative combinations are shown. 

CaseII(Ai>0, B, <0)-Case II {A,>0, B,<0): 



1327 


Casen{A,>0, <Q) - Case TV {A, <0, <0): 


1 

1 


1 

11 . 

1 

1 1 . 


■ + 




- 

•^2,. 






-F^ F" 
^1/ •*■2/ 

C* 

+ 


F2 1 

rC" = 



[T^ _r3 

L “ 1/ ^ 21 




2/J 


r,rj 


where, F sr and T 2 r have been defined in Eqns. (45c, d), and A^- and Br are given by 

O’" 

G 


(39a') 

(39b') 

(46a') 

(46b*) 



If the rotating shaft is strain-free, then r can be reduced to simple forms representing typical 
boundary conditions such as simple support, clamped support, and free end as shown in Ref. [6]. 
Note that for those supports in the strain-free case, the reflection matrices are constant. 


1328 


ANALYTICAL MODELLING OF COUPLED VIBRATIONS OF 
ELASTICALLY SUPPORTED CHANNELS 

Yavuz YAMAN 

Department of Aeronautical Engineering, Middle East Technical University 
06531 Ankara, Turkey 

An exact analytical method is presented for the analysis of forced vibrations 
of uniform thickness, open-section channels which are elastically supported at 
their ends. The centroids and the shear centers of the channel cross-sections do 
not coincide; hence the flexural and the torsional vibrations are coupled. Ends 
of the channels are constrained with springs which provide finite transverse, 
rotational and torsional stiffnesses. During the analysis, excitation is taken in 
the form of a point harmonic force and the channels are assumed to be of type 
Euler-Bernoulli beam with St.Venant torsion and torsional warping stiffness. 
The study uses the wave propagation approach in constructing the analytical 
model. Both uncoupled and double coupling analyses are performed. Various 
response and mode shape curves are presented. 

1. INTRODUCTION 

Open-section channels are widely used in aeronautical structures as stiffeners. 
These are usually made of beams in which the centroids of the cross-section 
and the shear centers do not coincide. This, inevitably leads to the coupling of 
possible flexural and torsional vibrations. If the channels are symmetric with 
respect to an axis, the flexural vibrations in one direction and the torsional 
vibrations are coupled. The flexural vibrations in mutually perpendicular 
direction occur independently. In the context of this study, this type of 
coupling is referred to as double-coupling. If there is no cross-sectional 
symmetry, all the flexural and torsional vibrations are coupled. This is called 
as triple-coupling. The coupling mechanism alters the otherwise uncoupled 
response characteristics of the structure to a great extent. 

This problem have intrigued the scientists for long time. Gere et al [1], Lin 
[2], Dokumaci [3] and Bishop et al [4] developed exact analytical models for 
the determination of coupled vibration characteristics. All those works, though 
pioneering in nature, basically aimed to determine the free vibration 
characteristics of open-section channels. 

The method proposed by Cremer et al [5] allowed the determination of 
forced vibration characteristics, provided that the structure is uniform in 
cross-section. The use of that method was found to be extremely useful when 
the responses of uniform structures to point harmonic forces or line harmonic 
loads were calculated. Mead and Yaman presented analytical models for the 


1329 


analysis of forced vibrations of Euler-Bernoulli beams [6]. In that they 
considered finite length beams , being periodic or non-periodic, and studied 
the effects of various classical or non-classical boundary conditions on the 
flexural response. Yaman in [7] developed mathematical models for the 
analysis of the infinite and periodic beams, periodic or non-periodic Kirchoff 
plates and three-layered, highly damped sandwich plates. 

Yaman in [8] also developed analytical models for the coupled vibration 
analysis of doubly and triply coupled channels having classical end boundary 
conditions. In that the coupled vibration characteristics are expressed in 
terms of the coupled wave numbers of the structures. The structures are first 
assumed to be infinite in length, and hence the displacements due to external 
forcing(s) are formulated. The displacements due to the waves reflected from 
the ends of the finite structure are also separately determined. Through the 
superposition of these two, a displacement field is proposed. The application 
of the end boundary conditions gives the unknowns of the model. The 
analytical method yields a matrix equation of unknowns which is to be solved 
numerically. The order of the matrix equation varies depending on the number 
of coupled waves. If the cross-section is symmetric with respect to an axis 
( double-coupling) and if the warping constraint is neglected, the order is six. 
If there is no cross-sectional symmetry (triple-coupling) and if one also 
includes the effects of warping constraint, the order then becomes twelve. 
This order is independent of the number of externally applied point forces. 
Although the method is basically intended to calculate the forced response 
characteristics, it conveniently allows the computation of free vibration 
characteristics as well. The velocity or acceleration of a point can easily be 
found. The mode shapes can also be determined. Both undamped and damped 
analyses can be undertaken. 

This study is based on the models developed in reference [8] and aims to 
analyze the effects of non-classical end boundary conditions on the coupled 
vibratory responses. If the ends are elastically supported (which may also have 
inertial properties) the problem becomes so tedious to tackle through the 
means of classical analytical approaches. The current method alleviates the 
difficulties encountered in the consideration of complex end boundary 
conditions. 


In this study a typical channel, assumed to be of type Euler-Bernoulli beam, is 
analyzed. It represents the double-coupling. Effects of the elastic end 
boundary conditions on the resonance frequencies, response levels and mode 
shapes are analyzed. Characteristics of otherwise uncoupled vibrations are 
also shown. 


1330 


2. THEORY 


2.1 Flexural Wave Propagation in Uniform Euler-Bernoulli Beams 

Consider a uniform Euler-Bernoulli beam of length L which is subjected to a 
harmonically varying point force Fo e acting at x=Xf. The total flexural 
displacement of the beam at any Xr (0 < Xr < L) can be found to be [5-8], 


w(x,,t) = ( i;A„e''„\ + Fo i ane'^' 


r r 


)e' 


( 1 ) 


The first series of the equation represents the effects of four waves which are 
being reflected from the ends of the finite beam. They are called free-waves. 
The second series accounts for the waves which are being created by the 
application of the external force Fo e on the infinite beam. Those waves are 
known as forced-waves, kn is the n’th wave number of the beam and 
kn =(mco^/EI)'^'‘ where m= Mass per unit length of the beam, co= Angular 
frequency, EI= Flexural rigidity of the beam, an values are the complex 
coefficients which are to be found by satisfying the relevant compatibiliy and 
continuity conditions at the point of application of the harmonic force [6,7]. 
An values, on the other hand are the complex amplitudes of the free waves and 
are found by satisfying the required boundary conditions at the ends of the 
beam. Once determined, their substitution to equation (1) yields the flexural 
displacement at any point on the finite beam due to a transversely applied 
point harmonic force. More comprehensive information can be found in [7]. 

2.2 Torsional Wave Propagation in Uniform Bars 

If one requires to determine the torsional displacements generated by a point, 
harmonically varying torque, a similar approach to the one given in Section 
2.1 can be used. In that case, the total torsional displacement can be written as: 


= )e‘”' (2) 

k is the wave number of the purely torsional wave and is known to be 
k=(-pIoa)VGJ)^^^ . k 2 = -ki and GJ=Torsional rigidity of the beam, p=Material 
density, Io=Polar second moment of area of the cross-section with respect to 
the shear centre. Toe'“ ^ is the external harmonic torque applied at x=Xt and 
b=l/(2kGJ). Bn values are the complex amplitudes of the torsional free-waves 
and are found by satisfying the appropriate end torsional boundary conditions. 

The consideration of the warping constraint To modifies equation (2) to the 
following form. 


1331 


( 3 ) 


«x„t) = (£C„e^\ + Toic„e-'=„'V,' ) e 

n*l »•! 

Now kn are the roots of 


EFo kn'^-GJkn^-pIoCO^=0 


( 4 ) 


Cn values are found by satisfying the necessary equilibrium and compatibility 
conditions at the point of application of the point harmonic torque acting on an 
infinite bar [8]. Cn values are determined from the end torsional boundary 
conditions of the finite bar. 

2.3 End Boundary Conditions for Uncoupled Vibrations 


2.3.1 Purely Flexural Vibrations 

Consider an Euler-Bernoulli beam of length L which is supported by springs at 
its ends. The springs provide finite transverse and rotational constraints Kt and 
Kr respectively. The elastic end boundary conditions can be foimd to be: 

El w”(0) - Kr I w’(0)=0 El w’”(0) + Ktj w(0)=0 

El w”(L) + K r,r w’(L)=0 El w’”(L) - Kt,rW(L)=0 (5) 


Here w’= 


dw(x) 


w 


d‘w(x) 


and w’” 


dV(x) 


w(x) is the spatially 


dx ’ dx“ dx' 

dependent part of equation (1) and second subscripts 1 and r allows one to use 
different stiffnesses for left and right ends. A more comprehensive study on 
these aspects can be found in references [6,7]. 


2.3.2 Purely Torsional Vibrations 

Now consider a bar of length L which is supported by torsional springs, 
having finite Ktor,at its ends. The elastic end boundary conditions requires that, 

Torque (0) - Ktor,i (|)(0)=0 and Torque (L) + Ktor .r (i)(L)=0 (6) 

Depending on the consideration of the warping constraint To, the torque has 
the following forms 

Torque(x)=GJ —— or Torque(x)= GJ —— - Ei o , 3 (/) 

dx dx ux 


1332 


(})(x) in equation (7) should be obtained either from equation (2) or equation (3) 
depending on the warping constraint r©. 

2.4 Doubly-Coupled Vibrations 

Now, consider Figure 1 . It defines a typical open cross-section which is 
synunetric with respect to y axis 


V z 



(b) 

Figure 1 : A Typical Cross-section of Double-coupling 

( a. Coordinate System, b. Real and Effective Loadings 
C: Centroid, O: Shear Centre) 

A transverse load applied through C results in a transverse load through O and 
a twisting torque about O. In this case the flexural vibrations in z direction are 
coupled with the torsional vibrations whereas the flexural vibrations in y 
direction occur independently. The motion equation of the coupled vibrations 
is known to be [1,2]. 


a-w a-(j) 


= 0 


a‘‘(i) 3^6 a’w a"(j) 


( 8 ) 


If one assumes that, 


w(x,t) = w„eVe”' 

(|)(x.t) = 4>„eVe“’ (9) 


1333 


Then, it can be found that, a load Pz through the centroid will create the 
following displacements at any x (0 < x < L) along the length of the 
channel [8], 

w(x,t) = (2A„e^’‘+Pjt a„ e'^ )e'"' 

n^l nvl 

n«l n>I 

Now kn values are the coupled wave numbers, An values are the complex 
amplitudes of the coupled free waves, an values are the complex coefficients 
which are to be found by satisfying the required compatibility and continuity 
conditions and 'Pn^ ( (El^ kn'^-mco^) / (CymcD^)) [8]. 

If required, the warping displacement u(x,t) can be found from (l)(x,t )as 

u(x,t)=-2A,^iM (11) 

dx 

where As is the swept area. 

Here 2j gives the order of the motion equation. j=3 defines the case in which 
the effects of warping constraint are neglected and j=4 represents the case 
which includes the warping effects. 


An values are found by satisfying the necessary 2j end boundary conditions. If 
warping constraint is neglected, the required six boundary conditions have the 
general forms given in equations (5) and (6). But the forms of w(x) and (j)(x) 
are now those given by equations (10) with j=3. If the warping constraint is 
included in the analysis the boundary conditions become eight. The six of 
those are again found by considering equations (10) with j=4 and substituting 
the resultant forms into equations (5) and (6). The remaining two can be 
found by evaluating equation (11) at both ends. 

When the flexural and torsional displacement expressions are substituted into 
the relevant equations, a set of equations is obtained. For the case of a load 
Pz and no warping constraint, the following equations can be found for j=3. 

EI^ w’”(0) + Kt,iw(0)=0 

EI^ (E kn^ An+ (-1) PzZ - kn^ a „ 6 ' "f ' ) 

^t,l ( E ■^n ■^PzE^n^ n f ) “0 (12) 


1334 


(13) 


EI^w”(0) -Kr,i w’(0)=0: 

EI^( I kn'An + Pz t kn'an 6 ‘ ' ) 

n«l n«t 

■ ( S kn An + Pz^'kn^n® n f ) 

n*l n“l 


GJ^^l«=o-K,„,.i W)=0: 

dx 

GJ( S k„ T„ A„ + (-1) (Pz cy) t -k„ 'f „ a„ e ' ’‘f') 

fl«l 11'I 

-K,or,i ( i % A„ + (Pz Cy) i >?„ a„ e ')=0 


diKx) I 

GJ-^Ix.l + K,„„ <KL)=0: 
dx 

GJ( 2k„>P„ A„ e“+ (PzCy) i-k„'P„a„ ) 

ns I 11=1 

+ K,„,.r (I % A„ + (Pz Cy) t Tn a„ e "f ')=0 


El5W”(L)+ Kr,rW’(L)=0: 

El^CS k„^A„e^‘-+(Pz) i k„^a„ ) 

n-l ii’*) 

+ Kz,r( i k„A„ e“+ (Pz) t-k„a„ e■^'‘-^')=0 

n»l n=i 


(14) 


(15) 


(16) 


EI^ w’”(L)-Kt,rW(L)=0: 

Eiaz kn'Ane'n^ +(Pz) 2 - kn^ a „ e n '^ ) 

11=1 Iia] 

-K,|(Z A„eV+(Pz) Za„e^''-V)=0 (17) 

lt=l 11=1 

Here (-1) multipliers are included due to the symmetry and anti-symmetry 
effects. 

Those equations can be cast into the following matrix form. 


= -{Terms containing Pz} (18) 



1335 


An eighth order equation represents the necessary matrix equation for the 
determination of An values if the warping constraint is included in the analysis. 
In that case, equations (12), (13), (16) and (17) are valid with j=4. On the other 
hand equations (14) and (15) should be replaced by, 

(GJ^-Er„^^)L.o <l>(0)=0: 

dx dx 

(GJ( £ k„ % A„ + (-1) (P. Cy) £ -k„ a„ e * „' ’‘f') - 

nol n»l 

Er„( £ k„3 'f „ A„ + (- 1 ) (P^ Cy) £ -k„ = % a„ e * „ ' -f')) 

it«l IIS’! 

-K,„,,,(|;'P„A„+(P,Cy)i 'P„a„e-^'’‘f')=0 (19) 

cr A/T^=A. 

(GJ “ EFo j 3 )• x=L I^tor>r y(L) 0 . 
dx dx 

(GJ( Jkn'Fn A„ e’=„‘-+ (PjCy) i-kn'Pnan ' )- 

n=t Ii»l 

EFoC Xkn^ 'J'n A„ eV+ (PzCy) £-k„^'P„a„ ')) 

11=11 n-l 

+ Kior,, ( X % A„ + (Pz Cy) X % a„ e * „' \ ')=0 (20) 

n-] ii3| 

where j=4. The remaining two equations are found by considering the warping 
of the extreme ends. If the ends are free to warp the axial stress is zero, if the 
ends are not to warp the axial displacements are zero at both ends. No elastic 
constraints are imposed on end warping. If the left end is free to warp and the 
right end is not to warp, the required boundary conditions can be shown to be; 

u’(0)=0:(x kn^'Pn A„ + PzX 'i'.ian k„^e-^ ' ^') = 0 ( 21 ) 

u(L)=0 :(X k„ e^‘' % A„ + PzX (-k„) '‘-"f') = 0 (22) 

n=] list 

All the equations can be put into the following matrix form 

A.' 

A: 

A, 

• = -{Terms containing Pz} (23) 

A, 

A. 

A. 

A. 



1336 


Required An values are numerically found from equations (18) or (23). Their 
substitution to the appropriate forms of equations (10) and (11) yield the 
required responses at any point on the beam. 

3.RESULTS AND DISCUSSION 

The theoretical model used in the study is shown in Figure 1 and has the 
following geometric and material properties: 

L=l(m), A =1.0*10"'(m\ h = 5.0*10-^(m), 15 = 4.17*10-* (m\ 

Cy = 15.625*10'* (m), J = 3.33*lO"'' (m*), =7.26*10'* (m"'), p= 2700 (kg/m*), 
r„ =2.85*10''* (m*), E = 7*10'“ (N/m*), G = 2.6* 10'“ (N/m*). 

Structural damping for torsional vibrations is included through, complex 
torsional rigidity as GJ*=GJ(l+z|3). For coupled vibrations, it is also included 
through the complex flexural rigidity as El^(H-z'n). 

First presented are the results for purely torsional vibrations. A bar assumed to 
have the given L, p, lo, G and J values is considered. The bar is then restrained 
at both ends by springs having the same torsional stiffness Ktor- A very low 
damping, p=10'^, is assigned and the resonance frequencies are precisely 
determined. It is found that, the introduction of a small Ktor introduces a very 
low valued resonance frequency. That fundamental frequency increases with 
increasing Ktor and as torsional constraint reaches to very high values, it 
approaches to the fundamental natural frequency of torsionally fixed-fixed 
beam. Table 1 gives the fundamental frequencies for a range of Ktor values and 
Figure 2 represents the fundamental mode shapes for selected Ktor values. 


Table 1: Uncoupled Fundamental Torsional Resonance Frequencies 

((3=10’^,No Warping Constraint) 


Ktor [N1 


Frequency [Hz] 


0 

0. 

10'^ 

1.606 

10'' 

5.035 

io“ 

14.683 

10' 

28.407 

10^ 

32.678 

10^ 

33.187 

10^ 

33.239 

;10*“ 

33.245 


1337 





Figure 6 on the other hand represents the low frequency torsional receptances 
of the case in which the warping constraint is taken into consideration and 
the ends are free to warp. This graph is included in order to show the variation 
of fundamental torsional resonance frequencies for a range of Ktor values. 



_- K,3r= 

K„,= 

kK,„= 


5 * 10 ': 

1 * 10 ': 

5 - 10 " 

1 - 10 ' 

5 - 10 ' 

1*10® 


Figure 6. Fundamental Frequencies of Purely Torsional Vibrations 
((3=0.01, x=0.13579[m],Warping Constraint Included, Ends are Free to Warp) 


The second part of the study investigates the characteristics of doubly-coupled 
vibrations. Now, the effects of each constraint are separately considered. A 
channel having the given parameters is supported at its ends by springs Kt, Kr, 
and Ktor- Warping constraint is included in the analysis and the ends are 
assumed to warp freely. First analyzed is the effects of Ktor- For this Kt=10^° 
[N/m] and Kr =10^° [N] are assigned at both ends of the channel and kept fixed 
throughout the study. Ktor is varied and the frequencies are shown in Table 2. 


Table 2. Effects of Ktnr in 

Doublv-counled Vibrations 

(Kt=10^° [N/m] and Kr=10^® [N], Warping Constraint Included) 

A: First Torsion Dominated Frequency [Hz] 

B: First Flexure Dominated Frequency [Hz] 

Ktor [Nl 

A 

B 

10-^ 

1.607 

134.603 

10'' 

5.069 

134.936 

10° 

15.620 

138.258 

10' 

39.725 

169.247 

10^ 

56.691 

318.423 

10^ 

59.528 

422.061 

10^ 

59.828 

430.642 


1338 








0.0 0.2 0.4 0.6 0.8 1.0 


ND LENGTH 

Figure 2. Fundamental Mode Shapes of Purely Torsional Vibrations 
((3=0, No Warping Constraint) 

Then, the warping constraint To is included in the analysis and the results of 
purely torsional vibrations are presented again. The beam had the same Ktor 
values at both ends and the numerical values of the relevant parameters are 
taken to be those previously defined. Figure 3 represents the fundamental 
mode shapes for which the ends are free to warp, whereas Figure 4 shows the 
mode shapes of the case in which there is no warping at the ends. 



Figure 3. Fundamental Mode Shapes of Purely Torsional Vibrations 


(P=0, Warping Constraint Included, Ends are Free to Warp) 






(P=0, Warping Constraint Included, Ends Can Not Warp) 


Figure 5 is drawn to highlight the effects of end warping. Both ends of the 
channel are restrained with Ktor=l * lO’ [N] and all the other parameters of the 
study are kept fixed. Figure 5 represents the direct torsional receptances of 
two cases in which the ends of the channel are allowed to warp and not to 
warp in turn. It can be seen that the prevention of end warping increases the 
resonant frequencies. 



Figure 5. Frequency Response of Purely Torsional Vibrations 

([3=0.01, X =0.13579 [m], Ktor=l*10' [N]» Warping Constraint Included) 


1340 



It can be seen that, when it has lower values Ktor is more effective on the 
torsion dominated resonance frequencies. For the higher Ktor values, the 
effects are more apparent on the flexure dominated frequencies. 

Figure 7 represents the direct flexural receptance of the channel for 
a set of selected end stiffnesses. Torsion dominated resonances at 59.528 [Hz], 
206.071 [Hz] and 476.649 [Hz] appear as spikes. The flexure dominated 
resonance occurs at 422.061 Hz. 



Figure 7. Frequency Response of Doubly-coupled Vibrations 
(ti=0.001, P=0.001, x=0.13579[m], Warping Constraint Included, 

Ends are free to warp, Kr=1.10^°[N], Kt=1.10^° [N/m], Ktor=l-10^ [N]) 

Then the effects of the rotational spring, Kr, are considered. The ends of the 
channel are assumed to be restrained with Kt=10^° [N/m] and Ktor =10^° [N]. 
The resulting frequencies are given in Table 3 for a range of Kr values. 


Table 3. Effects of Kr in Doublv-coupled Vibrations 

(Kt=10^° [N/m] and Ktor=10^° [N], Warping Constraint Included) 
A; First Torsion Dominated Frequency [Hz] 

B: First Flexure Dominated Frequency [Hz]_ 


Kr[N] _A_ B 


10*^ 

58.678 

205.344 

10° 

58.679 

205.358 

10^ 

58.699 

206.688 

10^ 

58.858 

218.055 

10" 

59.437 

289.738 

10^ 

59.799 

405.667 

10^ 

59.855 

429.345 


1341 



It is seen that Kr is not effective on torsion dominated resonance frequencies, 
but plays significant role for flexure dominated resonance frequencies. 

Finally considered the effects of the transverse spring Kt. Again, the channel is 
assumed to have very high Kr and Ktor values at both ends and Kt values are 
varied. Table 4 shows the resonance frequencies. 


Table 4. Effects of Kt in Doublv-coupled Vibrations 


(Kr=10^° [N] and Ktor=10^° [N], Warping Constraint Included) 
A: First Torsion Dominated Frequency [Hz] 

B: First Flexure Dominated Frequency [Hz]_ 


Kt [N/m] 

A 

B 

10' 

1.369 

70.455 

10^ 

4.328 

70.501 

10^ 

13.592 

70.973 

10^ 

39.282 

77.379 

10= 

58.095 

159.665 

10= 

59.699 

361.255 

10* 

59.860 

431.215 


It can be seen that the transverse stiffness, like torsional stiffness, effects both 
flexure and torsion dominated frequencies. 


4.CONCLUSIONS 


In this study, a new analytical method is presented for the analysis of forced 
vibrations of open section channels in which the flexible supports provide the 
end constraints. The dynamic response of open section channels is a coupled 
problem and their analysis requires the simultaneous consideration of all the 
possible vibratory motions. The wave propagation approach is an efficient tool 
for this complicated problem and the developed method is based on that. 

The current method analyzes the forced, coupled vibrations of open section 
channels. The channels, taken as Euler-Bernoulli beams, have uniform cross- 
section and a single symmetry axis. That consecutively leads to the coupling 
of flexural vibrations in one direction and torsional vibrations. The excitation 
is assumed to be in the form of a harmonic point force, acting at the centroid. 


1342 







Various frequency response curves of uncoupled and coupled vibrations are 
presented for a variety of different elastic end boundary conditions( which may 
also have the inertial properties). The developed method, although aimed at 
determining the forced vibration characteristics, is also capable of determining 
the free vibration properties. This is also demonstrated by presenting various 
mode shape graphs. It has been determined that the transverse and the torsional 
stiffnesses play more significant role as compared to the rotational stifness. 

The method can be used in analyzing the effects of multi point and/or 
distributed loadings. This can simply be achieved by modifying the terms of 
the forcing vector without increasing the order of the relevant matrix equation. 
The developed method can also be used in the analysis of elastically 
supported, triply-coupled vibrations of uniform channels. Results of that study 
will be the subject of another paper. 


REFERENCES 

1. Gere, J.M. and Lin, Y.K., Coupled Vibrations of Thin-Walled Beams of 
Open Cross-Section. J. AppliedMech Trans.ASME.,\9SZ, 80,373-8. 

2. Lin, Y.K., Coupled Vibrations of Restrained Thin-Walled Beams. 

J. Applied Mech. Trans.ASME., 1960, 82, 739-40. 

3. Dokumaci, E., An Exact Solution for Coupled Bending and Torsional 
Vibrations of Uniform Beams Having Single Cross-Sectional Symmetry. 
JSoundandVib.Am, 119,443-9. 

4. Bishop, R.E.D, Cannon, S.M. and Miao, S., On Coupled Bending and 
Torsional Vibration of Uniform Beams. J.Sound and Fi'/).,1989,131,457-64. 

5. Cremer, L. and Heckl, y\..,Structure~ Borne Sound, Springer-Verlag,1988. 

6. Mead, D.J. and Yaman, Y., The Harmonic Response of Uniform Beams on 
Multiple Linear Supports: A Flexural Wave Analysis. J. Sound and Vib, 
1990, 141,465-84 

7. Yaman, Y. Wave Receptance Analysis of Vibrating Beams and Stiffened 
Plates. PA Z). Ttew, University of Southampton, 1989. 

8 Yaman, Y., Vibrations of Open-Section Channels: A Coupled Flexural and 
Torsional Wave Analysis. (J. Sound and Vib, Accepted for publication) 


1343 



1344 


THE RESPONSE OF TWO-DIMENSIONAL PERIODIC STRUCTURES 
TO HARMONIC AND IMPULSIVE POINT LOADING 


R.S. Langley 

Department of Aeronautics and Astronautics 
University of Southampton 
Southampton S017 IBJ 


ABSTRACT 

Much previous work has appeared on the response of a two-dimensional 
periodic structure to distributed loading, such as that arising from a harmonic 
pressure wave. In contrast the present work is concerned with the response 
of a periodic structure to localised forcing, and specifically the response of the 
system to both harmonic and impulsive point loading is considered by 
employing the method of stationary phase. It is shown that the response can 
display a complex spatial pattern which could potentially be exploited to 
reduce the level of vibration transmitted to sensitive equipment. 

1. INTRODUCTION 

Many types of engineering structure are of a repetitive or periodic 
construction, in the sense that the basic design consists of a structural unit 
which is repeated in a regular pattern, at least over certain regions of the 
structure. An orthogonally stiffened plate or shell forms one example of an 
ideal two-dimensional periodic structure in which the fundamental structural 
unit is an edge stiffened panel. Although a completely periodic structure is 
unlikely to occur in practice, much can be ascertained regarding the structural 
dynamic properties of a real structure by considering the behaviour of a 
suitable periodic idealization. For this reason, much previous work has been 
performed on the dynamic behaviour of two-dimensional periodic structures, 
with particular emphasis on free vibration and the response to pressure wave 
excitation [1,2]. However, no results have yet been appeared regarding the 
response of two-dimensional periodic structures to point loading (as might 
arise from equipment mounts), and this topic forms the subject of the present 
work. A general method of computing the response to both harmonic and 
impulsive loading is presented, and this is then applied to an example system. 

Initially the response of a two-dimensional periodic structure to harmonic 
point loading is considered, and it is shown that the far-field response can be 
expressed very simply in terms of the "phase constant" surfaces which 
describe the propagation of plane waves. It is further shown that for 


1345 


excitation within a pass band two distinct forms of response can occur; in the 
first case the amplitude of the response has a fairly smooth spatial distribution, 
whereas in the second case a very uneven distribution is obtained and "shadow 
zones" of very low response are obtained. The second form of behaviour is 
related to the occurrence of caustics (defined in section 3.3), and the 
distinctive nature of the response suggests that a periodic structure might be 
designed to act as a spatial filter to isolate sensitive equipment from an 
excitation source. 

Attention is then turned to the impulse response of a two-dimensional periodic 
structure. It is again shown that the response can be expressed in terms of the 
phase constant surfaces which describe the propagation of plane waves. The 
application of the method of stationary phase to this problem has a number of 
interesting features, the most notable being the fact that four or more 
stationary points can arise. It is found that a surface plot of the maximum 
response amplitude against spatial position reveals features which resemble the 
"caustic" distributions obtained under harmonic loading. 

2. RESPONSE TO A HARMONIC POINT LOAD 


2.1 Modal Formulation and Extension to the Infinite System 
A two-dimensional periodic structure consists of a basic unit which is repeated 
in two directions to form a regular pattern, as shown schematically in Figure 
1. Each unit shown in this figure might represent for example an edge 
stiffened curved panel in an aircraft fuselage structure, a three-dimensional 
beam assembly in a roof truss structure, or a pair of strings in the form of a 
"-f" in a cable net structure. The displacement w of the system can be 
written in the form w{n,x), where n={n^ nf) identifies a particular unit and 
x = {Xi X 2 X 3 ) identifies a particular point within the unit. The coordinate 
system x is taken to be local to each unit, and the precise dimension of both 
X and the response vector w will depend on the details of the system under 
consideration. 

The present section is concerned with the response of a two- 
dimensional periodic structure to harmonic point loading of frequency co. In 
the case of a system of finite dimension, the response at location (n,x) to a 
harmonic force F applied at (0 ,jCo) can be expressed in the standard form [3] 


»’(n,*)=EE 

P 9 




( 1 ) 


where rj is the loss factor, 4>p^(n,x) are the modes of vibration of the system 
and (j}pg are the associated natural frequencies. The modes ^p^ which appear 
in equation ( 1 ) are scaled to unit generalized mass, so that 


1346 


/!, «! V 


( 2 ) 


where V represents the volume (or equivalent) of a unit and p(jc) is the mass 
density. The present concern is with the response of an infinite system, or 
equivalently the response of a large finite system in which the vibration decays 
to a negligible level before reaching the system boundaries. In this case the 
response is independent of the system boundary conditions, and it follows that 
any analytically convenient set of modes can be employed in equation (1). As 
explained in reference [4], it is expedient to consider the Born-Von Karman 
(or "periodic”) boundary conditions, as in this case the modes of vibration can 
be expressed very simply in terms of propagating plane wave components. 
In this regard it can be noted from periodic structure theory [5] that a 
propagating plane wave of frequency w has the general form 

w(rt,jc)=i?^{^(jc)exp(zej«, +ie^n^nu)t)}, (3) 


where and eo are known as the propagation constants of the wave (with - 
T<e,<T and - 7 r<G 2 <T for uniqueness), and g{x) is a complex amplitude 
function. By considering the dynamics of a single unit of the system and 
applying Bloch’s Theorem [5], it is possible to derive a dispersion equation 
which must be satisfied by the triad (w, 61 , 62 ) - by specifying Gj and €2 this 
equation can be solved to yield the admissible propagation frequencies w. By 
way of example, solutions yielded by this procedure for a plate which rests on 
a grillage of simple supports are shown in Figure 2 (after reference [ 6 ]). It 
is clear that the solutions form surfaces above the 61-62 plane - these surfaces 
are usually referred to as "phase constant" surfaces, and a single surface will 
be represented here by the equation a;= 0 ( 61 , 62 ). The phase constant surfaces 
always have cyclic symmetry of order two, so that 0 (ei, 62 )= 0 (- 6 i,- 62 ); for an 
orthotropic system the surfaces also have cyclic symmetry of order four, and 
therefore only the first quadrant of the 61-62 plane need be considered 
explicitly, as in Figure 2. 

The key point about the Born-Von Karman boundary conditions is that 
a single propagating wave can fully satisfy these conditions providing and 
6 o are chosen appropriately. The conditions state that the left hand edge of the 
system is contiguous with the right hand edge, and similarly the top edge is 
contiguous with the bottom edge, so that the system behaves as if it were 
topologically equivalent to a torus. If the system is comprised of XN 2 
units, then a propagating wave will satisfy these conditions if and 

62 ^ 2 =2x^ for any integers p and q. Following equation (3), the displacement 
associated with such a wave can be written in the form 


1347 




( 4 ) 


where ei^ and € 2 , are the appropriate values of the phase constants, and 
<^^^=0(ej^,e2g)- Now since it follows that a wave of 

frequency travelling in the opposite direction to will also satisfy the 
boundary conditions. This wave say) will have the form 

where it has been noted from periodic structure theory that reversing the 
direction of a wave leads to the conjugate of the complex amplitude function 
^(x). The two waves represented by equations (4) and (5) can be combined 
with the appropriate phase to produce two modes of vibration of the system 
in the form 


KSn,x) 

• 

Re 



Im 




( 6 ) 


By adopting this set of modes it can be shown [4] that equation (1) can be re¬ 
expressed as 

K.(«,x)= £ £ 2g;WF^g„(xJexp(-ie,^«,-%,n,) 

^=1-^/2 9=1 -Nj/2 +Z?7) -( J ? 


where and M have been taken to be even, and the amplitude function gp^ 
is scaled so that 

g^,=[2p(x)WV,Ar,]-%(x), {llV)\^f„(x)f;,(x)dx=\, (8,9) 

where the normalized amplitude function fp^ is defined accordingly. The 
summation which appears in equation (7) includes only those modes associated 
with a single phase constant surface 0 (€i, 62 ); if more than one surface occurs 
then the equation should be summed over the complete set of surfaces. The 
summation will include modes for each surface, which is consistent with 
known results for the modal density of a two-dimensional periodic structure. 

Equation (7) yields the response of a finite system of dimension XM 
to a harmonic point load - this response is identical to that of an infinite 
system if the vibration decays to a negligible amount before meeting the 
system boundaries. If the system size is allowed to tend to infinity in equation 


1348 


(7) then neighbouring values of the phase constants e^p and € 2 ^ become closely 
spaced (since deip=ei_p+i-eip=2T/A^i and de 2 g=€ 2 , 9 + 2 x 77 / 2 )> and in this 
case the summations can be replaced by integrals over the phase constants to 
yield 


- LI me„e,)ninv)-c^^ 


where 0 )= 0 ( 61 , 62 ) and ^(x) is the complex amplitude associated with the wave 
(£ 0 , 61 , 62 ). The evaluation of the integrals which appear in equation (10) is 
discussed in the following sub-sections. 


2.2 Integration over 61 

The integral over 6 i which appears in equation (10) can be evaluated by using 
contour integration techniques. Two possible contours in the complex e, plane 
are shown in Figure 3; to ensure a zero contribution from the segment 
Im( 6 i) = ±oo, the upper contour is appropriate for /Zi <0 while the lower 
contour should be used for n^X). For each contour the contributions from 
the segments and 61 ;;=x cancel, since the integrand which appears in 

equation (10) is unchanged by an increment of 2x in the real part of ei. The 
only non-zero contribution to the integral around either contour therefore 
arises from the segment which lies along the real axis. The poles of the 
integrand occur at the 61 solutions of the equation 

[fl(6i,62)?(U/i7)-a;^=0, (11) 

for specified 62 and oj. By definition there will be two real solutions^ in the 
absence of damping ( 77 = 0 ) providing the frequency range covered by the 
phase constant surface includes oj. Any complex solutions to equation (11) in 
the absence of damping will correspond physically to "evanescent" waves 
which decay rapidly away from the applied load. The present analysis is 
concerned primarily with the response of the system in the far field (that is, 
at points remote from the excitation source), and for this reason attention is 
focused solely on those roots to equation (11) which are real when 77 = 0 . The 
effect of damping on these roots can readily be deduced: if 77 is small then it 
follows from equation ( 11 ) that a real solution 6 ^ will be modified to become 
6 I-i(o 7 / 2 )( 5 Q/^ 6 l)■^ and hence the real pole for which dn/ 36 i <0 is moved to 
the upper half plane, while that for which 30/56, >0 is moved to the lower 
half plane. Given that the residue at such a pole is proportional to (30/36i)'\ 


^One positive and one negative. These solutions will have the form ± 6 , 
for an orthotropic system. 


1349 


it follows that the sign of the residue which arises from the contour integral 
is determined by the integration path selected, and hence by the sign of 
These considerations lead to the result 


~Tr~l fi|3fi/3«,|(l+/)/) 


( 12 ) 


where e,(£ 2 ,M) is the appropriate solution to equation (11). The evaluation of 
the integral over €3 is discussed in the following section. 

3.3 Integration over €3 

Since the present concern is with the response of the system at some distance 
from the excitation point, the integral over e, which appears in equation ( 12 ) 
can be evaluated to an acceptable degree of accuracy by using the method of 
steepest descent [7], With this approach it is first necessary to identify the 
value of €2 for which the exponent -i(eirti+e 2 « 2 ) is stationary. The condition 
for this is 

{3ejde^n^+n^=0. (13) 


Now Gi and satisfy the dispersion relation, equation (11), and thus equation 
(13) can be re-expressed in the form 

(aQ/a62)«r(5^2/a€,)«2=o, (i4) 


where it has been noted from equation ( 11 ) that, for fixed co, 3 ei/ 3 € 2 =- 
( 5 Q/ 3 e 2 )/( 9 fi/ 36 i), In the absence of damping the wave group velocity lies in 
the direction (SQ/Sei and in this case it follows from equation (14) 

that the group velocity associated with the required value of €2 is along («i ru). 
For light damping this result will be substantially unaltered, although damping 
will have an important effect on the value of the exponent -\{€^n^-\-e 2 iv^ at the 
stationary point. This effect can be investigated by noting initially that 

d{e^n^+&^n^)IBr}~{deJbri B&Jbr]).{n^ n^. (15) 

Now it follows from equation (11) that for light damping {ri<l) 

(dQ/de, dQlde,).{de,ldr] de^/dv) = -io)l2, (16) 


and hence equations (14)-(16) can be combined to yield the following result 
at the stationary point 

d{e^n^+e^n2)/dr) = -io)n/2c^. (17) 


1350 


Here Cg=^{{d£l!dex?+{d^lbe^'^] is the resultant group velocity and 
n=V[ni+ni-'\ is the radial distance (in units) from the excitation point to the 
unit under consideration. It follows that in the immediate vicinity of the 
stationary point the exponent can be expanded in the form 


(18) 

where the subscript 0 indicates that the term is to be evaluated at the 
stationary point under the condition 7 ]= 0 ; for ease of notation, this subscript 
is omitted in the following analysis. The method of steepest descent proceeds 
by substituting equation (18) into equation ( 12 ) and assuming that: (i) the main 
contribution to the integral arises from values of 62 in the immediate vicinity 
of the stationary point; (ii) the integrand is effectively constant in this vicinity, 
other than through variation of the term e 2 -(e 2 )o which appears in equation 
(18); (iii) under conditions (i) and (ii) the integration range can be extended 
to an infinite path without significantly altering the result. The method then 
yields [7] 

w{n,x) = ~if *F 7 o[ 20 V|aQ/a£j/ 2 xp(A:)p(A:o)|«i(aV 5 e 2 )| . 


(19) 

where/is the normalized complex wave amplitude which appears in equation 
(9), and all terms are to be evaluated at the stationary point. 

The stationary point associated with equation (19) is that point for 
which the group velocity is in the {n^ rQ direction. Geometrically, this is the 
point at which the normal to the curve a)=Q(ei, £ 2 ) itt the plane lies in the 
{n^ Ho) direction. Three such curves are shown schematically in Figure 4, 
together with a specified (n, 722 ) direction. For the frequencies and coj the 
situation is straight forward, in the sense that a unique stationary point exists 
for any {n^ n^) direction. For the frequency 0 J 2 the situation is more complex, 
since: (i) two stationary points occur for the (n^ rQ direction shown, and (ii) 
no stationary point exists if the {n^ direction lies beyond the heading B 
shown in the figure (the dashed arrow represents the normal with maximum 
inclination to the axis). In case (i) equation (19) should be summed over 
the two stationary points, while in case (ii) the method of steepest descent 
predicts that w{n,x) will be approximately zero, leading to a region of very 
low vibrational response. If the direction («i coincides with the dashed 
arrow, then equation (19) breaks down, since it can be shown that 
at this point. The heading indicated by the dashed arrow represents a caustic 
[7], and the theory given in the present section must be modified for headings 


1351 



(Wi Wj) which are in the immediate vicinity of the caustic - full details of the 
appropriate modifications are given in reference [4]. An example which 
illustrates the application of equation (19) is given in section 4. 

3. RESPONSE TO AN IMPULSIVE POINT LOAD 

If the system is subjected to an impulsive (i.e. a delta function applied at 
r=0), rather than harmonic, point load, then equation (10) becomes [8] 


w{n,x,t)={N^NJ2Tp) f [ g* 

II (20) 


where co = ^](€i,€ 2 )- The method of stationary phase can be applied to this 
expression to yield [8] 

w{n,x,t)-{\l2yr)[p{x)p{x)\J\r‘^^^^ 

exp(-/Gjtti -k^n^nQt+ib )}, 


( 21 ) 

where all terms are evaluated at the stationary point, and J and 5 are defined 
as 


/=(a"n/a6?)(a"Q/ae^-(3"0/ae,a6,)", S=(Tr/4)sgn(a%/fle?){l+sgn(/)}. 

(22,23) 

In this case the stationary point is given by the solution to the equations 

=(afi =(30 /3e,>. (24,25) 

In practice equations (24) and (25) may yield multiple solutions (stationary 
points), in which case equation (21) should be summed over all such points. 
Furthermore, stationary points having 7=0 indicate the occurrence of a 
caustic, and equation (21) must be modified in the immediate vicinity of such 
points as detailed in reference [8]. An example of the application of equation 
(21) is given in the following section. 


1352 



4. EXAMPLE APPLICATION 


4.1 The System Considered 

The foregoing analysis is applied in this section to a two-dimensional periodic 
structure which consists of a rectangular grid of lumped masses m which are 
coupled through horizontal and vertical shear springs of stiffness and h 
respectively. Each mass has a single degree of freedom consisting of the out- 
of-plane displacement w, and a linear spring of stiffness k is attached between 
each mass and a fixed base. It is readily shown that the system has a single 
phase constant surface of the form 

Q -(ep€,)=iLii(l-cos€i)+/x2(l-cos€2)+a;^, 


where ix^^lkjm, and o)^-==klm. The function U can be used in 

conjunction with the analysis of the previous sections to yield the response of 
the system to harmonic and impulsive point loading; in this regard it can be 
noted that for the present case p{x)=m, V—l, and/(:»:) = 1. 

4.2 Response to Harmonic Loading 

The surface is shown as a contour plot in Figure 5 for the case 

m=1.0, oj„==0, Ati = 1.0, Results for the forced harmonic response 

of this system at the two frequencies w = 1.003 and cx> = 1.181 are shown in 
Figures 6 and 7. In each case the response of a 40 x40 array of point masses 
is shown; a unit harmonic point load is applied to mass (21,21) and the loss 
factor is taken to be =0.05. Two sets of contours are shown in each Figure: 
the smooth contours have been calculated by using equation (19) while the 
more irregular contours have been obtained by a direct solution of the 
equations of motion of the finite 1600 degree-of-freedom system. By 
considering the results shown in Figure 6, it can be concluded that: (i) for the 
present level of damping the finite system effectively behaves like an infinite 
system, and (ii) the analytical result yielded by equation (19) provides a very 
good quantitative estimate of the far field response. It can be noted from 
Figure 5 that no caustic occurs for a) = 1.003, in the sense that equation (14) 
yields only one stationary point which contributes to equation (19). In 
contrast, a caustic does occur for the case a) = 1.181, and this leads to the very 
irregular spatial distribution of response shown in Figure 7. Two stationary 
points contribute to equation (19), and constructive and destructive 
interference between these contributions is responsible for the rapid 
fluctuations in the response amplitude. It is clear that the response exhibits 
a "dead zone" for points which lie beyond the caustic heading (in this case 
30.25° to the «i-axis), as predicted by the analysis presented in section 2. 

4.3 Response to Impulsive Loading 

The impulse response of a system having m=1.0, /xi = 1.0, ^2=0.51, and 


1353 


co„'“0.25 has been computed. The impulse was taken to act at the location 
/Zi=« 2=0 and the time history of the motion of each mass in the region - 
iO<(72i,n2) ^ 10 was found by using equations (21)-(23). For each mass the 
maximum response \w\ was recorded, and the results obtained are shown as 
a contour plot in Figure 8. In accordance with Fourier’s Theorem, the 
impulse response of the system contains contributions from all frequencies, 
and therefore the spatial distribution of | w| can be expected to lie somewhere 
between the two extreme forms of harmonic response exhibited in Figures 6 
and 7. This is in fact the case, and the response shown in Figure 8 retains a 
distinctive spatial pattern. As discussed in reference [8], the results shown in 
Figure 8 are in good agreement with direct simulation of the impulse response 
of the system. 


5. CONCLUSIONS 

This paper has considered the response of a two-dimensional periodic structure 
to both harmonic and impulsive point loading. With regard to harmonic 
loading, it has been shown that the spatial pattern of the response is strongly 
dependent on the occurrence of a caustic: if no caustic occurs then the 
response has a fairly smooth spatial distribution, whereas the presence of a 
caustic leads to an irregular spatial distribution and a "dead zone" of very low 
response. This type of feature is also exhibited, although to a lesser degree, 
in the spatial distribution of the response to an impulsive point load. This 
behaviour could possibly be exploited to reduce vibration transmission along 
a specified path, although the practicality of this approach for a complex 
system has yet to be investigated. The present analytical approach can be 
applied to all types of two-dimensional periodic structure - the information 
required consists of the phase constant surface(s) 0 ( 61 , 62 ) and the associated 
wave form(s) f{x), both of which are yielded by standard techniques for the 
analysis of free wave motion in periodic structures [1,2]. 

REFERENCES 

1. S.S. MESTER and H. BENAROYA 1995 Shock and Vibration 2, 69- 
95. Periodic and near-periodic structures. 

2. D.J. MEAD 1996 Journal of Sound and Vibration 190, 495-524. 
Wave propagation in continuous periodic structures: research 
contributions from Southampton 1964-1995. 

3. L. MEIROVITCH 1986 Elements of Vibration Analysis, Second 
Edition. New York: McGraw-Hill Book Company. 

4. R.S. LANGLEY 1996 Journal of Sound and Vibration (to appear). 
The response of two-dimensional periodic structures to point harmonic 
forcing. 


1354 





1 









■ 







■ 

■ 

1 

■ 

■ 

■ 

0 




m 

■ 

1 


■ 






Xj. ■j' 




Oi 




<- 






^1 













1 ^ 



Figure I, Schematic of a two-dimensional periodic structure. The arrow indicates' the reference unit 
(with n=0) while the circle represents a general point (re,x). The structure may have a third spatial 
coordinate X 3 , which for convenience is not shown in the figure. The point load considered in section 
3 is applied at the location of the arrow. 



Figure 2. Phase constant surfaces for a plate which rests on a square grillage of simple supports. 
Q is a non-dimensional frequency which is def as Q=a)LV(m/D), where m and D are respectively 
the mass per unit area and the flexural rigidii^ the plate, and L is the support spacing. 


1355 











Figure 5. Contour plot of the phase constant surface for the case ^=0.57. The contours 

are separated by an increment Aco=0.1477. The two contours considered in section 4.2 are indicated 
as follows: (a) a) = 1.033; (b) a) = 1.181. 



Figure 6. Response |H'(n,x)|2 of the 40x4v .uass/spring system to a unit harmonic force of 
frequency w-1.033 applied at the location i=y=21. The contours correspond to the response levels 
lK«,x)!'=0.01, 0.02, andO.05. 


1357 










STICK-SLIP MOTION OF AN ELASTIC SLIDER SYSTEM ON 
A VIBRATING DISC 


HOuyang J E Mottershead M P CartmeU' MIFiiswell^ 
Department of Mechanical Engineering, University of Liverpool 
^ Department of Mechanical Engineering, University of Edinburgh 
^ Department of Mechanical Engineering, University of Wales Swansea 

ABSTRACT 


The in-plane vibration of a slider-mass which is driven around the surface of a 
flexible disc, and the transverse vibration of the disc, are investigated. The disc 
is taken to be an elastic annular plate and the slider has flexibility in the 
circumferential (in-plane) and transverse directions. The static fiiction 
coefi&cient is assumed to be higher than the kinetic friction. As a result of the 
fiiction force acting between the disc and the slider system, the slider will 
oscillate in the stick-slip mode in the plane of the disc. The transverse vibration 
induced by the slider will change the normal force of the slider system acting 
on the disc, which in turn will change the in-plane oscillation of the slider. For 
different values of system parameters, the coupled in-plane oscillation of the 
slider and transverse vibration of the disc will exhibit quasi-periodic as well as 
chaotic behaviour. Rich patterns of chaotic vibration of the slider system are 
presented in graphs to illustrate the special behaviour of this non-smooth 
nonlinear dynamical system The motivation of this work is to analyse and 
understand the instability and/or squeal of physical systems such as car brake 
discs where there are vibrations induced by non-smooth dry-fiiction forces. 


NOMENCLATURE 


a , b mner and outer radii of the annular disc 
c damping coefficient of the slider in in-plane direction 
h thickness of the disc 


i = V=T 


1359 



k, k transverse and in-plane stiffiiess of the slider system 


m 

r 


t 


^stiek 


mass of the slider 

radial co-ordinate in cylindrical co-ordinate system 
radial position of the slider 

modal co-ordinate for k nodal circles and / nodal diameters 
time 

the time of the onset of sticking 
u , Mp transverse displacement of the slider mass and its initial value 
w , Wq transverse displacement of the disc and its initial value 
flexural rigidity of the disc 
Kelvin-type damping coefficient 
Young’s modulus 

initial normal load on the disc jfrom the slider system 
total normal force on the disc from the slider system 
combination of Bessel functions representing mode shape in radial 
direction 

circumferential co-ordinate of cylindrical co-ordinate system 

kinetic and static dry fiction coefficient between the shder and the 


D 

D' 

E 

N 

P 

R. 


0 


disc 


V 

P 

(P 

^ stick 
¥ 
¥ki 


CO. 


Poisson’s ratio 

damping ratio of the disc 

specific density of the disc 

absolute circumferential position of the slider 

absolute circumferential position of the slider when it sticks to the disc 
circumferential position of the slider relative to the drive point 
mode function for the transverse vibration of the disc corresponding to 

^kl 

natural (circular) frequency correq)onding to < 5 ^^ 


1360 


Q constant rotating speed of the drive point around the disc in radians per 
second 


INTRODUCTION 


There exists a whole class of mechanical systems which involve discs rotating 
relative to stationery parts, such as car brake discs, clutches, saws, computer 
discs and so on. In these systems, dry-friction induced vibration plays a crucial 
role in system performance. If the vibration becomes excessive, the system 
might fail, or cease to perform properly, or make offensive noises. In this 
paper, we investigate the vibration of an m-plane slider system, with a 
transverse mass-spring-damper, attached through an in-plane spring to a drive 
point which rotates at constant speed around an elastic disc, and the vibrations 
of the disc. Dry friction acts between the sHder system and the disc. 

Dry-friction induced vibration has been studied extensively [1-4]. For car brake 
vibration and squeal, see the review papers [5,6]. The stick-slip phenomenon of 
dry-friction induced vibration is studied in the context of chaotic vibration [7- 
10]. Popp and Stelter [7] studied such motion of one and two degrees of 
freedom system and foimd chaos and bifiircation. They also conducted 
experiments on a beam and a circular plate (infinite number of degrees of 
freedom). These theoretical works are about systems of less than three degrees 
of freedom, and the carrier which activates the friction is assumed to be rigid. 
In this paper, we consider an elastic disc so that the transverse vibrations of the 
disc are important. As a result of including the transverse vibrations of the disc, 
rich patterns of chaos, which have not been reported previously are found. If 
there is only shding present at constant speed, the problem is reduced to a 
linear parametric analysis which was carried out for a pin-on-disc system in 
[11] and for a pad-on-disc system in [12,13]. 

m-PLANE OSCILLATION OF THE SLIDER SYSTEM 


As the drive point, which is connected to the shder-mass through an in-plane, 
elastic spring, is rotated at constant angular speed around the disc, the driven 
slider will undergo stick-slip oscillations. The whole system of the shder and 
the disc is shown in Figure 1. 

The equation of the in-plane motion of the slider system relative to the rotating 
drive point, in the sliding phase, is. 


1361 


( 1 ) 


while in sticking, the equation of the motion becomes. 

The relationship between the relative motion of the slider system to the drive 
point and its absolute motion (relative to the stationary disc) is 

(p = Qt + y/, (3) 

We consider the foEowing initial conditions which are intended to simulate 
what happens in a disc brake. The slider system is at rest and there is no normal 
loading on the disc j&om the slider. Then a constant normal load is applied 
which causes transverse vibrations in the disc. At the same time, the drive point 
is given a constant angular velocity. Other initial conditions are possible, so 
that there is no loss of generality. 

First, sliding from the initial sticking phase occurs when, 

The slider will stick to the disc agaiu when, 

ju^P (during sliding), (5) 

or it will begin to sHde again iJ^ 

ij/ = , \k^rQtp-\ > (dxiring sticking). (6) 

Consequently, the slider system will stick and slide consecutively on the disc 
surface. 


TRANSVERSE VIBRATION OF THE ANNULAR DISC 

The equation of motion of the disc under the slider system is, 

ph^ + D'V^'w + DW = --5(r-r„)5i$-<p)P. (7) 

a/" r 

The total force P is the summation of initial normal load N and the resultant 
of the transverse motion w of the slider. Its expression is, 

P - N+ mu + cu-hkiu-u^). (8) 

Since it is assumed that the slider system is always in perfect contact with the 
disc, then, 


1362 



(9) 


u{t) = w{r„(p{t)A- 

Substitution of ecpiations (8) and (9) into (7) leads to. 


ph 


^ — + /:)VV = --5(r - r, )6(^ - (p)[N + 


dt 


dt 


^..dw d^w d^w .dw dw 


( 10 ) 


k(w-wj\. 

Note that equation (10) is valid whether the slider system is sticking or sliding. 
When the slider sticks to the disc, equation (10) reduces to. 


+ D-V‘ ^+£)W = --5(r -)5(5 -<p)x 

St St r (11) 

d'^W ^ M 

[A^+m—+ c—+ A:(w-w„)]. 


COUPLED VIBRATIONS OF THE SLIDER AND THE DISC 


Assume that the transverse motion of the disc can be represented by, 

M>{r,e,t)=ttii/,{r,0)q,it), ( 12 ) 

Jt=0 /=-<io 


and, 

where {r) is a combination of Bessel functions satisfying the boundary 
conditions in radial direction at the inner radius and outer radius of the disc. 
The modal functions satisfy the ortho-normality conditions ofj 




(14) 


1363 


Equations (10) and (11) can be simplified by being written in terms of the 
modal co-ordinates from equation (12). 

During sticking, the motion of the whole system of the slider and the disc can 
be represented by, 

N 

+ 2^0)- 
—Z^„('-o)^«('-o)exp[i(5-0(»]x (15) 

ph.U ''=0 s=-«o 

{mq„+cq^+k{q^-q„{^)\). 


The sticking phase can be maintained i^ 


rA\¥\</^XJ^+- 


^Z Z ^«(?')exp(i/?»)x 


{mq„+cq„+ k[q„ - (0)]} ]. 

While in sliding, the motion of the whole system can be represented by. 


%+2^a>^,qu+a>lq„ = 


^^«('i)exp(-i/<»)- 


-rrri Z^„('-o)^«(n,)exp[i(5-0«!’]x 

pnO r=<i 5 ^=-oo 

H?,. +i2sw„ +isw„) + 

k{q„-qS'^)]}, 


r„{m{j/ + kw) = -/i,Sign((Z))[A'' +- 


^Z Z KMx 


exp(LS(Z>)M?„ +i2s^„ +(ls^-5>^)9„] + 

c{q„ +>s(?g„)+*[!?„ -?„(o)]}]. 

The sliding phase can be maintained if, 


1364 


\K^M<mXN+-J^ t t K{r,)es.p{is(p)x 
■yjphb ^=0^^ 

{iTiq„+i2sj>q„+(is^-s^j)^)qJ+ ( 20 ) 

c(?„ +isj^J + k[q„ -9„(0)]}], 

when, 

\j/ = ~0 or ^ = 0. (21) 

COMPUTING PROCESS 


As the shder system sticks and sHdes consecutively, the governing equations of 
the coupled motions of the whole system switch repeatedly from equations 
(15), (16) and (17) to equation (18), (19) and (20). TTie system is not smooth. 
Since the condition which controls the phases of the slider system itself 
depends on the motions, it is also a nonlinear system, whether is a constant 
or a function of relative speed ^. In order to get modal co-ordinates, we have 
to truncate the mfinite series in equation (12) to jfinite terms. Then numerical 
integration is used to solve equations (15), (18) and (19). Here a fourth order 
Rimge-Kutta method is used for second order simultaneous ordinary 
differential equations. 

Since equation (18) has time-dependent coefficients, time step length has to be 
very small. Constant time step lengths are chosen when the m-plane slider 
motion is well within the sticking phase or the shding phase in the numerical 
integration. As it is imperative that the time step should be chosen such that at 
the end of some time intervals the shder happens to be on the sticking-shding 
interfaces, we use a prediction criterion to choose next time step length when 
approaching these interfaces. Therefore, at the sticking-shding interfaces, the 
time step length is variable (actuahy smaher than it is while weh within sticking 
or shding). Nevertheless, tbe interfaces equations ( equation (17) or equations 
(20-21)) are only approximately satisfied [10]. 

When transverse motion of the shder system becomes so violent that the total 
normal force P in it becomes negative or becomes several times larger than 
the initial normal load 7/, we describe the system as being unstable. Then the 
motion begins to diverge. But this instabihty should be distinguished from a 
chaotic motion which is bounded but never converge to a point. 


1365 


NUMERICAL EXAMPLES 


The following data are used in the computation of numerical examples: 
a = 0.065m, b = 012m, OTm,/z = 0.001m; = 120GPa, v= 0.35, 

Z)*= 0.00004; yW, =0.4, //^=0.24, A: = lOOON/m, = lOON/m, 

m = 0.1kg, p = 7000kg/m^ The disc is clamped at inner radius and free at 

outer radius. Note that in these numerical examples, the disc thickness is 
dehherately taken to be very small in order to reduce the amount of computing 
work. However, this will not affect the qualitative features of the results or 
conclusion drawn from the results thus obtained. The first five natural 
(circular) frequencies are 451.29, 462.73, 426.73, 508.23, 508.23. We will 
concentrate on the vibration solutions at different levels of initial normal load. 
But occasionally solutions at different rotating speed or different damping 
ratios are investigated. Unless specified expressly, the Poincare sections are for 
the in-plane vibration of the slider system 

First of all, we study the effect of the normal loadA^. Take f2=l0 and 
^— When N is very small^ the Poincare section is a perfect ellipse 

which indicate the in-plane vibration of the slider system is quasi-periodic, as 
the transverse vibration of the disc is too small to affect total normal force P . 
A typical plot of such motion is shown in Figure 2 for A^=0.5kPa. As N 
increases, the sticking period gets longer, the bottom part of the elhpse evolves 
into a straight line, indicating phase points within the sticking phase. One of 
such plots is given in Figure 3 for N =3kPa. A further increase of N not only 
lengthens the straight line part of the Poincare section, but also creates an 
increasingly ragged outline in the arch part of the plot. The curve is no longer 
smooth and it seems that the in-plane motion begins to enter a chaotic state 
from the quasi-periodic state. Figure 4 presents the Poincare section plot for 
N =7.5kPa. There is a transition period from quasi-periodic motion to chaotic 
motion, extending from N =6kPa up to N =9kPa. Chaos becomes detectable 
at iV=10kPa, whose Poincare section is shown in Figure 5. Then chaotic 
vibration follows. When N ~15kPa, the arch part of the Poincare section 
becomes so fuzzy and thick that it should no longer be considered as a curve, 
but rather a narrow (fractal) area. A hlow-up’ view of the arch part reveals 
that phase points are distributed across the arch. Both plots are ^own in 
Figure 6. Between iV'=17.ikPa and 18.325kPa, the vibration of the slider 
enters a new stage, with Poincare sections looking like star clusters as 
illustrated in Figures 7 and 8. This kind of motions are rather extraordinary and 


1366 


have not been reported in other works on stick-shp motions with a rigid 
carrier. Afterwards, the ‘arch-door’ hke Poincare sections come back (see 
Figure 9). The difference from previous Poincare sections of lower N is that 
the new Poincare sections look like overlapping of earlier Poincare sections, 
which indicates a clear layered structure, as diown in Figure 10 and more 
obviously in the left hand side of Figure 11. At this stage, the vibration is very 
chaotic. To give the reader a better picture, the Poincare section of a fixed 
point on the disc at (= 0.1m and = 0), is also shown in the right hand side 
of Figure 11. The Poincare sections of the slider-mass and a point on the disc 
are also given in Figures 12-15. In Figure 12 for A^=30.5kPa, the vibration 
goes unstable. Here again, the Poincare sections have not been reported 
elsewhere. 

If disc damping is increased, vibration will become more regular, as shown in 
Figure 13. Comparing Figures 11 with 13, we see that increase of disc damping 
makes the vibration more concentrated though not always smaller. Unstable 
vibration can be stabilised with more disc damping, as seen from Figure 14. 

If there is no damping at aU, the resulting vibration due to dry fiiction will be 
unstable, even at very small normal load N . In Figure 15, the motion of the 
slider tends to run away in the tangential direction from the normal ellipse 
attractor, while the motion of the disc goes unbounded. 

Increasing the speed of the drive point seems to make vibration more chaotic 
and more unstable, as shown in Figures 16-18. At this stage, however, we are 
unable to make a definite conclusion on rotating speed as there might be 
intervals of regular motions and intervals of chaotic motion for . More 
numerical examples must be computed to draw a positive conclusion on this 
parameter. 

The correlation dimension is not a good measure of the vibration for the 
current problem because its values fluctuate in some numerical examples. This 
failure was perhaps first discovered in [7]. The reason can be either that the 
system is non-smooth, or that the system has multiple degrees of freedom, or 
both. Therefore, the correlation dimension or any other fractal dimensions is 
not presented in this paper. 


CONCLUSIONS 


In this paper, we studied the in-plane stick-slip vibration of a slider system with 
a transverse mass-spring-damper driven around an elastic disc through a spring 
from a constant speed drive point, and transverse vibrations of the disc. The 


1367 


whole system had been reduced to six degrees of freedom after simplification. 
From numerical examples computed so far, we can conclude that: 

1. Both vibrations are very complex as this is a multi-degree of freedom, non¬ 
smooth system Rich patterns of chaotic vibration are found. Some have not 
been reported elsewhere. 

2. For the normal pressure parameter, smaller values allow quasi-peiiodic 
solutions. Greater pressures result in chaotic motions. At certain large 
pressures, the vibrations become unstable. 

3. Disc dartping makes vibration more concentrated to smaller areas and when 
sufficiently large it can stabilise otherwise unstable vibration. 

4. An increase in the rotating speed can make the vibration more chaotic or 
more unstable. 

5. Correlation dimension is not a good measure of the vibration of this multi¬ 
degree of freedom, non-smooth dynamical system 

6. Much more investigation needs to be earned out in understanding and 
characterising the vibration of multi-degree of freedom, non-smooth dynamical 
systems. 


ACKNOWLEDGEMENT 

This research is supported by the Engineering and Physical Sciences Research 
Council (grant niunber J35177) and BBA Friction Ltd. 

REFERENCES 


1. Nakai, M., Chiba, Y. and Yokoi, M., Railway wheel squeal. Bulletin of 
JSME, 1984, 27, 301-8 

2. Lin, Y-Q and Wang Y-H, Stick-sHp vibration of drill strings. XEng.Ind., 
TransASME, 1991,113, 38-43 

3. Ferri, A. A. and Bindemann, A.C., Damping and vibration of beams with 
various types of fiictional support conditions. J. VibAcoust, TransASME^ 
1992, 114, 289-96 

4. Lee, A.C., Study of disc brake noise using multi-body mechanism with 
friction interface. In Friction-Induced Vibration, Chatter, Squeal, and 
Chaos, Ed. Ibrahim, KA. and Soom, A., DE-Vol. 49, ASME 1992, pp.99- 
105 


1368 


5. Ibrahim, R.A., Friction-induced vibration, chatter, squeal, and chaos. In 
Friction-Induced Vibration, Chatter, Squeal, and Chaos, Ed. Ibrahim, 
KA. and Soom, A, DE-VoL 49, ASME 1992, pp. 107-38 

6. Yang, S. and Gibson, R.F., Brake vibration and noise: reviews ,comments, 
and proposed considerations. Proceedings of the 14th Modal Analysis 
Conference, The Society of Experimental Mechanics, Inc., 1996, pp.l342- 
9 

7. Popp, K and Stelter, P., Stick-slip vibration and chaos. Phil. Trans. R Soc. 
Lond. A(1990), 332, 89-106 

8. Pfeiffer, F. and Majek, M., Stick-slip motion of turbine blade dampers. Phil. 
Trans. R Soc. Lond. A(1992), 338, 503-18 

9. Wojewoda, J., Kapitaniak, T., Barron, R. and Brindley, J., Complex 
behaviour of a quasiperiodically forced experimental system with dry 
friction. Chaos, Solitons and Fractals, 1993, 3, 35-46 

10. Wiercigroch, M., A note on the switch function for the stick-slip 
phenomenon. J.SoundVib., 1994,175, 700-4 

11. Chan, S.N., Mottershead, J.E. and Cartmell, M.P., Parametric resonances 
at subcritical speeds in discs with rotating frictional loads. Proc. Instn. 
Mech. Engrs, 1994, 208, 417-25 

12. Mottershead, J.E., Ouyang, R, Cartmell, M.P. and Friswell, M.I., 
Parametric Resonances in an annular disc, with a rotating system of 
distributed mass and elasticity; and the effects of friction and dan:q)ing. 
Proc. Royal Soc. Lond. A., 1997, 453, 1-19 

13. Ouyang, H., Mottershead, J.E., Friswell, M.l. and Cartmell, M.P., On the 
prediction of squeal in automotive brakes. Proceedings of the 14th Modal 
Analysis Conference, The Society of Experimental Mechanics, Inc., 1996, 
pp. 1009-16 



Figure 1. Slider system and disc in cylindrical co-ordinate system 


1369 





Figure 9. iV=21kPa 


Figure 10. iV=24kPa 














1374 


A Finite Element Time Domain 
Multi-Mode Method For Large Amplitude 
Free Vibration of Composite Plates 

Raymond Y. Y. Lee, Yucheng Shi and Chuh Mei 
Department of Aerospace Engineering 
Old Dominion University, Norfolk, VA 23529-0247 


Abstract 

This paper presents a time-domain modal formulation using the finite element method for 
large-amplitude firee vibrations of generally laminated thin composite rectangular plates. Accurate 
fi'equency ratios for fundamental as well as higher modes of composite plates at various maximum 
deflections can be determined. The selection of the proper initial conditions for periodic plate 
motions is presented. Isotropic beam and plate can be treated as special cases of the composite 
plate. Percentage of participation from each linear mode to the total plate deflection can be 
obtained, and thus an accurate frequency ratio using a minimum number of linear modes can be 
assured. Another advantage of the present finite element method is that the procedure for obtaining 
the modal equations of the general DujB5ng-type is simple when compared with the classical 
continuum Galerkin’s approach. Accurate frequency ratios for isotropic beams and plates, and 
composite plates at various amplitudes are presented. 


Introduction 

Large amplitude vibrations of beams and plates have interested many 
investigators [1] ever since the first approximate solutions for simply supported 
beams by Woinowsky-Kiieger [2] and for rectangular plates by Chu and Herrmann 
[3] were presented. Singh et al. [4] gave an excellent review of various formulation 
and assumptions , including the finite element method for large amplitude firee 
vibration of beams. Srirangaraja [5] recently presented two alternative solutions, 
based on the method of multiple scales (MMS) and the ultraspherical polynomial 
approximation (UPA) method, for the large amplitude firee vibration of a simply 
supported beam. The fi'equency ratios for the fundamental mode, ca/C0L> at the ratio 
of maximum beam deflection to radius of gyration of 5.0 (Wmax/r =5.0) are 3.3438 
and 3.0914, using the MMS and the UPA method, respectively. Eleven firequency 
ratios including nine firom reference [4] were also given (see Table 1 of reference 
[5]). It is rather surprising that the firequency ratio for the fundamental mode at 
Wn«x/r =5.0 for a simply supported beam varied in a such wide range: fi-om the 
lowest of 2.0310 to the highest of 3.3438, and with the elliptic function solution by 
Woinowsky-Kiieger [2] giving 2.3501. Similar wide spread exists for the vibration 
of plates. Rao et al [6] presented a finite element method for the large amplitude 
firee flexural vibration of unstiffened plates. For the simply supported square plate. 


1375 



frequency ratios from six different approaches were reported (see Table 1 of 
reference [6]). The frequency ratio at Wmax/h =1.0 varied from a low of 1.2967 to a 
high of 1.5314, with Chu and Herrmann’s analytical solution [3] at 1.4023. 

This paper presents a finite element time domain modal formulation for the large 
amplitude free vibration of composite plates. The formulation is an extension from 
the isotropic plates [7], and the determination of initial conditions for periodic 
motions was not employed in reference [7]. The convergence of the fundamental 
frequency ratio is investigated for a simply supported beam and a simply supported 
square plate with a varying number of finite elements and a varying number of 
linear modes. Accurate frequency ratios for fundamental and higher modes at 
various maximum deflections, and percentages of participation from various linear 
modes, are obtained for beams and composite plates. 


Formulation 

Strain-Displacement and Constitutive Relations 

The von Karman strain-displacement relations are applied. The strains at any 
point z through the thickness are the sum of membrane and change of curvature 
strain components: 


{e} = 








-w.xx 



V,y 

>+< 

/ 2 

> + z< 

-W,yy 



U,y+V,x^ 


W,yj 




= {Em} + {sb} + z{K} 


( 1 ) 


where and {eb}are the membrane strain components due to in-plane 

displacements u and v and the transverse deflection w, respectively. The stress 
resultants, membrane force {N} and bending moment {M}, are related to the strain 
components as follows: 



■[A] IB]' 
[B1 [Di; 



( 2 ) 


where [A] is the elastic extensional matrix, [D] is the flexural rigidity matrix, and 
[B] is the extension coupling matrix of the laminated plate. 

Element Displacements, Matrices and Equations 

Proceeding from this point, the displacements in equation (1) are approximated 
over a typical plate element , e.g. rectangular [8] or triangular [9], using the 
corresponding interpolation functions. The in-plane displacements and the linear 
strains are interpolated from nodal values by 


1376 




where [Hm] and [Bm] denote the displacement and strain interpolation matrices, 
respectively, and {Wm} is the in-plane nodal displacement vector. The transverse 
displacement, slopes and curvatures are interpolated from the nodal values by 

w = [Hb]{w^}, |^’^| = [G]{wb}, {K} = [B|,]{wb} (4a,b,c) 


where [HJ and [G] and [BJ denote the bending displacement, slope and curvature 
interpolation matrices, respectively, and {wb } denotes the nodal transverse 
displacements and its derivatives. Through the use of Hamilton’s principle, the 
equations of motion for a plate element undergoing large amplitude vibration may 
be written in the form 


[hb] 

0 



T [H] 

[kB]l 

r[klNm]+[klNB] [klbml 

,r[k2b] < 




[k»]J 

L ^'"*1 0 J ' 

[ 0 ( 



or 

[m]{ w} + m + [kl] + [k2] }{w} = 0 (5) 


where [m] and [k] are constant matrices representing the element mass and linear 
stiffiaess characteristics, respectively; Pel] and Pe2] are the first order and second 
order non-linear sfiffiiess matrices, respectively; PcInJ depends linearly on 
unknown membrane displacement ({Nm }= [A][Bni]{wni}); PcInb] depends linearly 
on the unknown transverse displacement ({NB}=[B][Bb]{Wb}); DelbnJ depends 
linearly on the unknown plate slopes and represents coupling between membrane 
and bending displacements; and [k2b] depends quadratically on the unknown plate 
slopes. 


System Equations 

After assembling the individual finite elements for the complete plate and 
applying the kinematic boundary conditions, the finite element system equations of 
motion for the large-amplitude free vibration of a thin laminated composite plate 
can be expressed as 

[M]{w}4- ([K]+ [K1(W)] + [K2(W)]){W} = 0 (6) 

where [M] and [K] are constant matrices and represent the system mass and 
stifiBness respectively; and [Kl] and [K2] are the first and second order nonlinear 
stifi&iess matrices and depend linearly and quadratically on the unknown structural 


1377 


nodal displacements {W}, respectively. Most of the finite element large amplitude 
fi-ee vibration results for plates and beams in the literature, e.g. references [1,6] and 
others, were based on eq. (6) using an iterative scheme and various approximate 
procedures. The system equations are not suitable for direct numerical integration 
because: a) the nonlinear stif&iess matrices [Kl] and [K2] are functions of the 
unknown nodal displacements, and b) the number of degrees of fireedom (DOF) of 
the system nodal displacements {W} is usually too large. Therefore, eq. (6) has to 
be transformed into modal or generalized coordinates followed by a reduction of 
the number of DOF. In addition, the general DufiSng-type modal equations will 
have constant nonlinear modal stifihiess matrices. This is accomplished by a modal 
transformation and truncation 


r=l 

where and cou are the natural mode (normalized with the maximum 

component to unity) and linear firequency from the eigen-solution ©|_j.[M]{ 
(|))«=[K]{ (|)}®. 

The nonlinear stififtiess matrices [Kl] and [K2] in eq. (6) can now be expressed 
as the sum of the products of modal coordinates and nonlinear modal stiffaess 
matrices as 


[Kl] = ^q,(t)[Kl((])«)] (8) 

r=I 

n n 

[K2] = ^ ^ qr (t)qs (t)[K2((|)(''), )] (9) 

r=l s=l 

The nonlinear modal stifihiess matrices [Kl]^"^ and [O]^'^^ are assembled from the 
element nonlinear modal stif&iess terms [kl]^^^ and Pc2]^“^ as 

([K1]«.[K2]‘“))= ^([klf>,[k2]'“>) (10) 

al! elements 
+ bdy. conds. 


where the element nonlinear modal stif&iess matrices are evaluated with the known 
linear mode Thus, the nonlinear modal stiffiiess [Kl]^^^ and [K2]^'®^ are constant 
matrices. Equation (6) is thus transformed to the general Duf6ng-type modal 
equations as 


1378 


( 11 ) 


[M]{q}+([K]+[KlJ+[K2„]){q} = 0 
where the modal mass and linear stifhiess matrices are diagonal 
([M],[K]) = [<I.f([M],[K])[<I.] 
and the quadratic and cubic terms are 


LI 

[Kl,]{q} = [<Df J^qjKl] 


1 ^) 


Vt=1 
n n 






(rs) 


r=:l s=l 


\Wk} 


( 12 ) 


(13) 


(14) 


AH modal matrices in eq. (11) are constant matrices. With given initial 
conditions, the modal coordinate responses {q} can be determined from eq. (11) 
using any direct numerical integration scheme such as the Runge-Kutta or 
Newmark-P method. Therefore, no updating of the vibration modes is needed [10]. 
For periodic plate osciHations have the same period T, the response of all modal 
coordinates should also have the same period T. Since the initial conditions wHl 
affect greatly the modal response, the determination of initial conditions for periodic 
plate osciHations is to relate each of the rest modal coordinates in powers of the 
dominated coordinate as 


arqi(t;IC) + brqi(t;IC) + Crq^(t;IC)+.= qr(t;IC), r = 2,3. ...n 

(15) 

where the 2k, br, Cr , . are constants to be determined, and IC denotes initial 

conditions. For a three-mode (n=3) system, it is accurate enough to keep up to the 
cubic term only in eq. (15) and this leads to two set of equations 

a2qi(tp;A,B,C) + b2qf(tp;A,B,C) + C2qJ(tp;A,B,C) = q2(tp;A,B,C), p = 1,2,3 
a3qi (tp;A, B,C) + bsqJ (tp; A, B. C) + CsqJ (tp;A. B, C) = q3 (tp;A, B, C), p = 1,2,3 

(16a,b) 

in which the modal coordinates qi, q 2 and qs at tp are known quantities and the 
initial conditions are qi(0)=A, q2(0) =B, q3(0)=C and qi(0) = q2(0) = q3(0) = 0. 
PracticaHy, only eight equations are needed to determine the eight unknowns a 2 , as. 


1379 





hi, b 3 , C 2 , C 3 , B and C through an iterative scheme. However, the number of 
equations can be more than the number of unknowns for accurate determination of 
initial conditions and the least square method is employed in this case. 

The time history of the plate maximum deflection can be obtained from eq. (7). 
The participation value from the r th linear mode to the total deflection is defined 
as 


maxjqj. 

n 

^max|qi 

i=l 


(17) 


Thus, the minimum number of the linear modes for an accurate and converged 
frequency solution can be determined based on the modal participation values. 


Results and Discussions 

Assessment of Single-Mode Elliptic Function Solution 

The fundamental frequency ratio Co/col = 2.3051 at Wmax/r =5.0 for a simply 
supported beam obtained by Woinowsky-Kiieger [2] using a single-mode and 
elliptic fimetion is assessed first. The conventional beam element having six (four 
bending and two axial) DOF is used. A half-beam is modeled with 10, 15, 20 
elements, and the lowest four symmetrical linear modes are used in the Duffing 
modal equations. Table 1 shows that a 20-element and 1-mode model will yield a 
converged result. The percentages of participation from each mode for various 
values of Wmax/r are given in Table 2. The modal participation values demonstrate 
that a single mode (n=l) will yield an accurate fundamental frequency because the 
contribution from higher linear modes to the total deflection is negligible (< 0.01 % 
for Wmax/r up to 5.0). There is a small difference in frequency ratios between the 
present finite element and the elliptic integral solutions. This is due to the difference 
between the axial forces of the two approaches, the finite element method (FEM) 
gives a non-uniform axial force in each element; however, the average value of the 
axial force for each element is the same as the one in the classic continuum 
approach. The lowest (2.0310) and the highest (3.3438) frequency ratios at Wmax/r 
=5.0 in reference [5] are not accurate. 

Frequency ratios for higher modes of the simply supported beam are obtained 
next. A model with 40-elements and 3-anti-symmetric modes for the whole beam is 
employed for the frequency ratio of the second nonlinear mode. The mode 
participations shown in Table 2 indicate that a single-mode approach will yield 
accurate frequency results. And the frequency ratios for the second mode are the 
same as those of the fundamental one. Thus, the present method agrees extremely 
well with Woinowsky-Krieger’s classic single-mode approach. 


1380 



The time history of the first two symmetric modal coordinates and the beam 
central displacement, phase plot, and power spectral density (PSD) at maximum 
beam deflection W^Jr = 5.0 for the fimdamental firequency (or mode) are shown in 
Fig. 1. The time scale is non-dimensional and Ti is the period of the fundamental 
linear resonance. It is noted that although the central displacement response looks 
like a simple harmonic motion, it does have a small deviation fi:om pure harmonic 
motion due to the second small peak in the spectrum. This is in agreement with 
classical solution that the ratio of the jfrequency of the second small peak to that of 
the first dominant peak is 3. 

Now we are ready to assess the single-mode fundamental firequency of a simply 
supported square plate obtained by Chu and Herrmann [3]. A quarter of the plate is 
modeled with 6 x 6, 7 x 7, 8 x 8 and 9 x 9 mesh sizes and 1, 2, 4 or 5 symmetrical 
modes are used. The conforming rectangular plate element with 24 (16 bending 
and 8 membrane) DOF is used. The in-plane boundary conditions are u = v = 0 on 
all four edges. Table 3 shows that the 8 x 8 mesh size in a quarter-plate and 4- 
mode model should be used for a converged and accurate frequency solution. Table 
4 shows the fi:equency ratios and modal participation values for the lowest three 
modes at various Wmax/h for a simply supported square plate (8x8 mesh size in a 
quarter-plate). It indicated that at least two linear modes are needed for an accurate 
frequency prediction at Wmax/h =1.0, and the contribution of higher linear modes 
increase with the increase of plate deflections. The modal participation values also 
show that the combined modes (1,3)-(3,1) and (2,4)-(4,2) are independent of the 
large-amplitude vibrations dominated by (1,1) and (2,2) modes, respectively. The 
time history, phase plot, and PSD at the maximum deflection Wmax/h =1.0 for the 
fundamental mode are shown in Fig. 2a, and Tu is the period of the fundamental 
linear resonance. There is one small peak in the spectrum and the frequency ratio of 
the second small peak to the first dominant one is 3. The low (1.2967) and the high 
(1.5314) frequency ratios at Wmax/h =1.0 given in reference [6] are not accurate. 

The influence of the initial conditions on periodic motion is demonstrated in Fig. 
2a and 2b. In Fig. 2a, the modal coordinates all have the same period, and the initial 
conditions are determined fi*om eq.(15). They are qii(0)/h=1.0, qi3+3i(0)/h= - 
0.0155, qi 3 . 3 i (0)/h = 0.0, q33(0)/h=0.000813, and qi 5 + 5 i( 0 )/h= 0.00011, and initial 
velocities are null, whereas in Fig. 2b, qii(0)/h=1.0 and all others are nuU. The 
modal coordinates do not have the same period. 

Clamped Beam 

It is thus curious to find out whether multiple-mode is required for the clamped 
beam. Convergence study of the fundamental firequency ratios at Wmax/r =3.0 and 
5.0 shown in Table 5 indicates that a 25-element (half-beam) and 4-mode model 
win yield accurate and converged results. The time history, phase plot and PSD at 
Wmax/r =5.0 are shown in Fig. 3. The modal participation values in Table 6 and the 
PSD in Fig. 3 confirm that at least two modes are needed for accurate firequency 
results. 


1381 


Symmetric Composite Plate 

A simply supported eight-layer symmetrically laminated (0/45/-45/90)s 
composite plate with an aspect ratio of 2 is investigated. The graphite/epoxy 
material properties are as follows; Young’s moduli Ei = 155 GPa, E 2 = 8.07 GPa, 
shear modulus Gn = 4.55GPa, Poisson’s ratio V 12 =0.22, and mass density p = 
1550 kg/m^ A 12 X 12 mesh is used to model the plate. The in-plane boundary 
conditions are fixed (u=v=0) at all four edges. The first seven linear modes are used 
as the modal coordinates. Table 7 gives the fundamental firequency ratios and mode 
participation values for the linear modes in increasing firequency order. The modal 
participation values indicate clearly that four modes are needed in predicting the 
nonlinear fi-equency, and other three of the seven are independent of the 
fundamental nonlinear mode. Figure 4 shows the time-history, phase plot, and PSD 
at Wmax/h =1.0. 


UNS YMMETRIC COMPOSITE PLATE 

A simply supported two-layer laminated (0/90) composite plate of 15 x 12 x 
0.048 in. (38 x 30 x 0.12 cm) is investigated. The graphite/epoxy material 
properties are the same as those of the symmetric composite plate. A12 x 12 mesh 
is used to model the plate. The in-plane boundary conditions are fixed at all four 
edges. The first four linear modes are used as the modal coordinates. Table 8 gives 
the fundamental firequency ratios and mode participation values for the linear modes 
in increasing fi-equency order. From the phase plot, the time histories and PSD 
shown in Fig. 5, it can be seen that the total displacement response has a non-zero 
mean (i.e. the positive and negative displacement amplitudes for all modal 
coordinates are not equal). The quasi-ellipse in the phase plot is not symmetrical 
about the vertical velocity-axis. In the PSD at Wmax/h =1.0, it is observed that there 
are four small peaks in the spectrum and the firequency ratios of the second, third, 
fourth and fifth peak to the first dominant one are 2, 3, 4 and 5, respectively. This 
observation indicates that the displacement response includes the 
superharmonances of orders 2, 3, 4, and 5. The curves, which the positive and 
negative displacement amplitudes are plotted against the fundamental firequency 
ratio, are also given in Fig. 5. The difference between the positive and negative 
amplitudes increases as the firequency ratio increasing. 


Conclusions 

A multimode time-domain formulation, based on the finite element method, is 
presented for nonlinear firee vibration of composite plates. The use of FEM enables 
the present formulation to deal with composite plates of complex geometries and 
boundary conditions, and the use of the modal coordinate transformation enables to 
reduce the number of ordinary nonlinear differential modal equations to a much 
smaller one. The present procedure is able to obtain the general Duffing-type modal 
equations easily. Initial conditions for all modal coordinates having the same time 


1382 


period are presented. The participation value of the linear mode to the nonlinear 
deflection is quantified ; they can clearly determine the minimum number of linear 
modes needed for accurate nonlinear frequency results. 

The present fundamental nonlinear fi-equency ratios have been compared with 
the single-mode solution obtained by Woinowsky-Kneger for simply supported 
beams and by Chu and Herrmann for simply supported square plates. The 
Woinowsky-Krieger’s single-mode solution is accurate. For all other solutions, 
however, two or more modes are needed. The nonlinear firequencies for 
symmetrically and unsymmetrically laminated rectangular composite plates are also 
obtained. The phase plot and power spectral density showed that nonlinear 
displacement responses are no longer harmonic, and multiple modes are required 
for isotropic clamped beams and isotropic and composite plates. 


References 

1. M. Sathyamoorthy 1987 Applied Mechanics Review 40, 1553-1561. Nonlinear 
vibration analysis of plates: A review and survey of current developments. 

2. S. Woinowsky-Kreger 1950 Journal of Applied Mechanics 17, 35-36. The effect of an 
axial force on the vibration of hinged bars. 

3. H. N. Chu and G. Herrmann 1956 Journal of Applied Mechanics 23, 523-540. 
Influence of large amplitudes on jfree flexural vibrations of rectangular elastic plates. 

4. G. Singh, A. k. Sharma and G. V. Rao 1990 Journal of Sound and Vibration 142, 77- 
85. Large amplitude free vibration of beams-discussion of various formulations and 
assumptions. 

5. H. R. Srirangaraja 1994 Journal of Sound and Vibration 175, 425-427. Nonlinear free 
vibrations of uniform beams. 

6. S. R. Rao, A. H. Sheikh and M. Mukhopadhyay 1993 Journal of the Acoustical 
Society of America 93 (6), 3250-3257. Large-amplitude finite element flexural 
vibration of plates/stiffened plates. 

7. Y. Shi and C. Mei 1996 Journal Sound and Vibration 193, 453-464. A finite element 
time domain modal formulation for large amplitude free vibration of beams and plates. 

8. K. Bogna:, R. L. Fox and L. A. Schmit 1966 Proceeding of Conference on Matrix 
Methods in Structural Mechanics, AFFDL-TR-66-80, Wright-Patterson Air Force 
Base, Ohio, October 1965, 397-444. The gena-ation of interelement compatible 
stiffness and mass matrix using the interpolation formulas. 

9. Teseller and T. J. R. Hughes 1985 Computer Methods in Applied Mechanics and 
Engineering 50, 71 -101. A three node Mindlin plate element with improved fransverse 
shear. 

10. A. K. Noor 1981 Composites and Structures, 13, 31-44. Recent advances in reduction 
methods for nonlinear problems. 


1383 


Table 1. Convergence of the fundamental frequency ratio at Wmax/r =5.0 for a 
simply supported beam 


No. of elements 
and 4 modes 

(C0/(0l)i 

No. of modes 
and 20 elements 

(G)/G)l)i 

10 

2.3537 

1 

2.3506 

15 

2.3511 

2 

2.3506 

20 

2.3506 

3 

2.3506 

-- 


4 

2.3506 


Table 2 The lowest two frequency ratios and the modal participations for a 
simply supported beam 


W^nax/r 

Elliptic 
integral [2] 
(CO/OOl)! 



FEM 



Modal 

Participation % 


(CO/COl)! 



__ 

0.2 

1.0038 

1.0038 

100.00 

0.000 

0.000 

0.4 

1.0150 

1.0149 

100.00 

0.000 

0.000 

0.6 

1.0331 

1.0331 

100.00 

0.000 

0.000 

0.8 

1.0580 

1.0581 

100.00 

0.000 

0.000 

1 

1.0892 

1.0892 

100.00 

0.000 

0.000 

2 

1.3178 

1.3179 

100.00 

0.002 

0.000 

3 

1.6257 

1.6258 

100.00 

0.004 

0.000 

4 

1.9760 

1.9761 

99.99 

0.005 

0.000 

5 

2.3501 

2.3506 

99.99 

0.009 

0.000 

W.„ax/r 

(co/g)l)2 

(C0/CDl)2 

q.2_ _ 

q4 

Q6 

0.2 

1.0038 

1.0038 

100.00 

0.000 

0.000 

0.4 

1.0150 

1.0149 

100.00 

0.000 

0.000 

0.6 

1.0331 

1.0332 

100.00 

0.000 

0.001 

0.8 

1.0580 

1.0582 

100.00 

0.000 

0.001 

1 

1.0892 

1.0893 

100.00 

0.000 

0.002 

2 

1.3178 

1.3181 

99.99 

0.000 

0.006 

3 

1.6257 

1.6260 

99.98 

0.000 

0.015 

4 

1.9760 

1.9768 

99.98 

0.000 

0.021 

5 

2.3501 

2.3512 

99.96 

0.000 

0.037 


Table 3. Convergence of the fundamental frequency ratios for a simply 
supported square plate (Poisson’s ratio=0.3) 


Mesh sizes 
and 

(CD/Qjn 

at 

W^,Jh 

No. of modes 
and 

(co/coUn 

at 

W^Jh 

4 modes 

1.0 

1.4 

8x8 mesh 

1.0 

1.4 


6x6 

1.4174 

1.7423 

1 


1.7028 

7x7 

1.4163 

1.7396 

2 

1.4169 

1,7433 

8x8 

1.4164 

1.7403 

4 

1.4164 

1.7403 

9x9 

1.4164 

1.7400 

5 

1.4163 

1.7401 













Table 4. The lowest three frequency ratios and the modal participations for a 
simply supported square plate (Poisson’s ratio=0.3) 



Elliptic 



FEM 




integral[3] 

(CO/ffiiJii 

(to/oiOii 

Modal 

Participation 

% 



qu 

qi3 -1- qsi 

qi3 - qsi 

q33 

qi5 -f. qsi 

0.2 

1.0195 

1.0195 

99.93 

0.07 

0.00 

0.00 

0.00 

0.4 

1.0757 

1.0765 

99.72 

0.27 

0.00 

0.01 

0.00 

0.6 

1.1625 

1.1658 

99.38 

0.59 

0.00 

0.02 

0.00 

0.8 

1.2734 

1.2796 

98.93 

1.02 

0.00 

0.05 

0.01 

1.0 

1.4024 

1.4163 

98.34 

1.57 

0.00 

0.08 

0.01 

1.2 

1.5448 

1.5659 

97.54 

2.30 

0.00 

0.15 

0.01 

1.4 

1,6933 

1.7401 

96.29 

3.42 

0.00 

0.27 

0.02 



(C0/®l)21 

q2i 

q23 

q4i 

q43 

... 

0.2 

N/A 

1.0243 

99.93 

0.06 

0.01 

0.00 

... 

0.4 

N/A 

1.0976 

99.50 

0.45 

0.03 

0.01 

— 

0.6 

N/A 

1.2072 

98.15 

1.28 

0.54 

0.02 

... 

0.8 

N/A 

1.3411 

97.54 

2.41 

0.00 

0.05 

— 

1.0 

N/A 

1.5126 

96.24 

3.69 

0.00 

0.08 

... 

1.2 

N/A 

1.6900 

94.90 

4.92 

0.03 

0.15 

... 

1.4 

N/A 

1.8952 

93.54 

6.01 

0.01 

0.44 

... 

y/raJh 


(C0/0)l)22 

q22 

q24 + q42 

q24 - q42 

q44 

... 

0.2 

N/A 

1.0245 

100.00 


0.00 

0.00 

... 

0.4 

N/A 

1.0751 

100.00 

0.00 

0.00 

0.00 

... 

0.6 

N/A 

1.1611 

99.99 

0.00 

0.00 

0.01 

... 

0.8 

N/A 

1.2806 

99.98 

0.01 

0.00 

0.01 

... 

1.0 

N/A 

1.4041 

99.93 

0.01 

0.00 

0.06 

— 

1.2 

N/A 

1.5551 

99.97 

0.01 

0.00 

0.01 

... 

1.4 

N/A 

1.7074 

99.98 

0.02 

0.00 

0.00 

— 


Table 5, Convergence of the fundamental frequency ratios for a clamped beam 

No. of 
elements 

and 4 modes 

(co/©l)i 

at 

W™/r 

No. of modes 
and 

(©/©lOi 

at 

Wn.ax/r 

3.0 

5.0 

25 elements 

3.0 

5.0 

10 

1.1751 

1.4046 

1 



15 

1.1740 

1.4009 

2 



20 

1.1732 

1.3999 

3 



25 

1.1731 

1.3996 

4 




1385 













Table 6 The fundamental frequency ratios and the modal participations for a 
clamped beam 



Elliptic 

integral 

(0)/(0l)i 



FEM 



(ffl/(0L)l 

Modal Participation 

% 


qi 

q3 

qs 

_ 


1.0222 

1.0222 

99.78 

0.20 

0.02 

0.00 

2.0 

1.0857 

1.0841 

99.33 

0.58 

0.08 

0.02 

3.0 

1.1831 

1.1731 

98.35 

1.44 

0.17 

0.04 

4.0 

1.3064 

1.2817 

97.37 

2.28 

0.29 

0.07 

5.0 

1.4488 

1.3996 

96.26 

3.22 

0.42 

0.11 

Table 7 The fundamental frequency ratios and the modal participations for a 


simply supported rectangular (0/45/-45/90)s composite plate (a/b=2) 


WmJh 




Modal 

Participation 

% 




(ffl/coOn 

qu 

qi2 

q2i 

qi3 

q22 

q23 

qsi 

0.2 

1.0408 

99.51 

0.00 

0.00 

0.41 

0.07 

0.00 

0.02 

0.4 

1.1490 

96.57 

0.00 

0.00 

3.01 

0.24 

0.00 

0.17 

0.6 

1.3484 

92.93 

0.00 

0.00 

4.55 

0.47 

0.00 

2.04 

0.8 

1.5241 

98.51 

0.00 

0.00 

0.53 

0.94 

0.00 

0.02 

1.0 

1.7190 

97.43 

0.00 

0.00 

2.39 

0.11 

0.00 

0.07 

1.2 

1.9258 

95.78 

0.00 

0.00 

3.57 

0.62 

0.00 

0.02 

1.4 

2.1409 

94.27 

0.00 

0.00 

4.84 

0.77 

0.00 

0.13 

Table 8 

The fundamental frequency ratios and the modal participations for a 
simply supported rectangular (0/90) composite plate 

W„,Jh 



Modal Participation 

% 





(CO/COiOn 

qii 


qi3 

qsi 


q33 

0.2 


1.0358 

97.83 


1.18 

0.82 


0.18 

0.4 


1.1432 

95.13 


2.25 

2.24 


0.38 

0.6 


1.2993 

94.53 


3.86 

1.18 


0.60 

0.8 


1.5432 

88.56 


4.36 

4.77 


2.31 

1.0 


1.7880 

89.15 


3.31 

5.06 


2.48 

1.2 


2.0142 

92.22 


2.89 

3.15 


1.74 

1.4 


2.2823 

92.01 


2.92 

2.92 


2.15 


1386 













0.5 1 1.5 2 -6 -3 

Time Ratio (tn‘1) 



4 6 8 10 12 
Frequency Ratio 


U 

Displacement 


Figure 1. Time histories, phase plot and PSD for the fundamental 
mode at =5.0 of a simply supported beam 








Power Spectrum Density Total OlsplacemonllThIckness qSSn'hJcknBSS q11/ThIckness 

p P P T* Thousandths o o o 




Time Ratio (tn"11) 



-0.4 -0.2 0 0.2 0.4 

Displacement 



Figure 2a. Time histories, phase plot and PSD for the fundamental 
mode at W^u/h =1.0 of a simply supported square plate 


1388 






Thousandths 












Displacement Spectrum Density Total Displacement/Thickness q22/Thickness q1 IH'hickrtess 



0 12 3 4 

Time Ratio {t/T11) 




0 12 3 4 



0 2 4 6 8 10 12 

Frequency Ratio 



0 12 3 4 


Time Ratio {tfTII) 



0 12 3 4 

Time Ratio (t/TII) 



Displacement 


Figure 4. Time histories, plot and PSD for the fundamental 
mode at Wmax/h =1.0 of a simply supported 
(0/45M5/90)s rectangular plate 


1391 



Power Specirum DonsUy Total Dlsplacemenin'hlckness qSin’hlckness q11/Thlckness 













NONLINEAR FORCED VIBRATION OF BEAMS BY THE 
HIERARCHICAL FINITE ELEMENT METHOD 

P. Ribeiro and M. Petyt 

Institute of Sound and Vibration Research, University of Southampton, Southampton 

S017 IBJ, UK 

Abstract: The hierarchical finite element (HFEM) and harmonic balance 
methods are used to derive the equations of motion of beams, in steady-state forced 
vibration with large amplitude displacements. These equations are solved by the 
Newton and continuation methods. The stability of the obtained solutions is 
investigated by studying the evolution of perturbations of the solutions. Additionally, a 
method that allows a quick examination of the stability of the solution is presented and 
applied. The convergence properties of the HFEM, the influence of the number of 
degrees of freedom and of in-plane displacements are discussed. The HFEM results are 
compared with experimental results. Symbolic computation is used in the derivation of 
the model. 


NOTATION 


A - extension coefficient 
b - width of the beam 
B - coupling coefficient 

[C] - damping matrix 
D - bending coefficient 

[D] - Jacobian of {F} 

|D| - determinant of [D] 

E - Young's modulus 

[E] - elastic matrix 

{f} - vector of out-of plane shape functions 
|F} - vec. of amplitudes of generalised forces 

- vector of generalised forces 

{F} - vector of dynamic forces 

{g} - vector of in-plane shape functions 

h - length of the finite elements 

h - thickness of the beam 

[Klb] - linear bending stiffiiess matrix 

[Kip] - linear stretching stiffiiess matrix 

[K2], [K3] and [K4] - components of 

nonlinear stiffiiess matrix 

[Knl] - nonlinear stiffiiess matrix 

L - length of the beam 

[M] - mass matrix 

[Mb] - bending mass matrix 
[Mp] - in-plane mass matrix 

[N] - matrix of shape functions 
[N'''(x)J - row matrix of out-of-pl. sh. f 

p j - number of in-plane shape functions 


pQ - number of out-of-plane s. funct. 

{qp} - in-plane displacement fimction 
{qw} - transverse displacement function 
r - radius of gyration 
t - time 

u - in-plane displacement 
Ur - generalised in-plane displacements 
w - transverse displacement 
Wr - generalised out-of-pl. displ. 

{wc}, (wj - coef of cosine and sine terms 
X - axial coordinate of the beam 

a - loss factor 
p - damping factor 

eg ,s§ - linear membrane and bending strains 
- geometrically nonlin. membrane strain 
{si}, {£ 2 }- linear and geom. nonl. strains 
5 W - virtual work of the external forces 

cx 

SWy - virtual work of the internal forces 
5W- - virtual work of the inertia forces 

in 

X - characteristic exponent 

p - mass density 

CO - angular fi:equency 

cOjj. - natural frequencies 

Ja>y - diagonal matrix of squares of 

natural firequencies 
- Viscous damping ratio 


1393 


1-INTRODUCTION 


In real systems, due to large amplitudes of the excitation, small stiffiiess or 
excitation with a frequency in the neighbourhood of resonance frequencies, vibrations 
with large amplitudes can occur. In this case, the linear theories may not allow a good 
representation of the dynamic characteristics of the system. 

A typical case study of vibrations in the nonlinear regime is the forced vibration 
of beams, with large displacements. Although a large amount of investigation has been 
carried out in this field [1, 2, 10, 18 and others], a method that would allow the 
inclusion of higher order mode contributions and damping, without increasing 
excessively the number of degrees of freedom (d.o.f) is desirable. The purpose of this 
work is to apply and investigate the advantages of a method that satisfies these 
conditions: the hierarchical finite element method (HFEM). 

In the HFEM, to achieve better approximations, higher order shape functions 
are added to the existing model. Convergence tends to be achieved with far fewer 
d.o.f than in the /z-version of the finite element method [4, 11]. The linear matrices 
possess the embedding property, meaning that the associated element matrices for a 
number of shape functions n=ni are always submatrices for n=n 2 , n 2 >ni. The existing 
nonlinear matrices of an approximation of lower order, ni, can be used in the derivation 
of the nonlinear, matrices of the improved approximation, n 2 . This makes the construction 
of a more accurate model, potentially quicker in the HFEM than in the /z-version. 

We are going to consider that the time variation of the solution may be 
expressed by harmonics and use the harmonic balance method (HBM). Compared with 
perturbation methods, the main advantages of the HBM are its simplicity, the fact that 
it is not restricted to weakly non-linear problems and, for smooth systems, the 
assurance of convergence to the exact solution [5]. 

In nonlinear vibrations, the frequency response curves can have multi-valued 
regions, turning and bifurcation points. In these regions, we are going to use a 
continuation method [8, 14], because, if the Newton method alone is applied, the 
solution will depend heavily on the initial guess and convergence is very difficult to 
achieve. 

Symbolic computation [15] will be utilised, allowing an easier and more 
accurate construction of the model. 


1394 


2 - MATHEMATICAL MODEL 


The beam is assumed to be elastic and isotropic, with thin uniform thickness h. The 
effects of transverse shear deformations and rotatory inertia are neglected. The transverse 
displacement is large compared with the beam thickness, but is very small compared with 
the length of the beam (w « L). The slopes are also very small; (w «1. 


The displacement components u and w may be expressed as the combination of 
the hierarchical polynomial shape functions, 


§lS2"-Spi ^ 

0 f,f,..-fn 


{d} = |;;) = [N]{q}.[N] = 

{q}"=[U, ... U,, W. ... Wj. (2.1) 

The hierarchical polynomials used in this study were derived from Rodrigue's form 
of Legendre polynomials [4]. Only one element was used to model the whole beam and 
only the shape functions that satisfy the boundary conditions were included in the model. 
Applying the theory of Bemoulli-Euler, expressing the strain; 


1 as 


{s} = 


(w^f /2 
0 


P 

_JOo 


'Oj ■ 


( 2 . 2 ) 

I 0 j L8 “j [oJ -- -- 

and equating the virtual work of the inertia forces (D’ Alembert principle) to the 
virtual work of the external and elastic restoring forces we obtain: 

{5q}^{F}-bJJ{5s,}%{5s,r)[E]({e,} + {s,})dL= {5q}^[M]{q}, (2.3) 

° ; A. D = £(1,2^)Ed2; [M] = phbJjN]^[N]dL. 


[E] = 


The stiffness matrices are defined by: 


b[{58,}^[E]{s,}dL={5q}"[Kl]{q}.bJj6eT[E]{8,}dL={6qnK2]{q}. 
b({5e3r[E]{8,}dL={6q}"[K3]{q}, b[{5e,}"[E]{84dL={5qnK4]{q}. (2.4) 

Considering only transverse forces, if P(t) represents a concentrated force acting 
at the point x=Xj and (x, t) represents a distributed force, the generalised forces are 
{F} = (£p,(x.t){N”(x)}dL + P,(t){N”(x.)}). (2.5) 

In real systems energy is dissipated; consequently, damping should be included 
in the present model. For a large variety of materials experimental investigations show 


1395 


that the energy dissipated per cycle is not dependent on the frequency and is 
proportional to the square of the amplitude of vibration [12, 13], The corresponding 
type of damping is called hysteretic. We will represent it by a matrix proportional to 
the mass matrix and frequency dependent: 

[C] = i[M] (2.6) 

Considering that damping in the beam results only from the action of the linear axial 
and bending strains, we have the following equations of motion; 





"0 


_K3 

K4jj 




(2.7) 


The in-plane inertia can be neglected for slender beams [3] and the damping 
contribution due to the axial stress is generally negligible compared to that due to the 
bending stress [12], With these approximations and because [K3] = 2[K2]^, ref [4], we 

can simplify the equations of motion to obtain: 

+J[M.]{qw} + [Kl.]{q„} + [Knl]{q„} = {F}, (2.8) 

[Knl] = [K4] - 2[K2 ]^[k1,]'‘[K2] . (2.9) 

To integrate exactly terms involving shape functions or its derivatives, present 
in the stiffriess and mass matrices, symbolic computation was employed, using the 
package Maple [15]. 

If the external excitation is harmonic and if initial conditions are such that no 
transient response exists, then {qw(t)} may be expressed, in a first approximation, as: 


{qw(t)}={wc}cos(©t)+{Ws}sin(a)t) (2.10) 

We are going to insert this equation into the equations of motion (2.8) and 
apply the HBM. This method can be easily implemented in a program written with the 
symbolic manipulator For that, one defines the command trign using Maple 

library of trigonometric functions, trig, in the following way; 
trign: =readlih('trig/reduce') [15, 17]. trign, thus defined, replaces all nonlinear 
trigonometric functions by linear ones^. With the command coeff one selects the terms 
in cos(ot) and sin(a 3 t). In this way, we obtain equations of motion of the form: 


1 For example cos^(Q)t) 


is replaced by ^cos(a)t) 


+ .icos(3cot) 


1396 




-CO' 


M, 0 
0 M, 




-3M, 


KL 




( 2 . 11 ) 


{Fj=|0Knl]{q4cos(cDt)dt=gKM.l + iKM.3){wJ+iK^^^ (2.12) 

{F,}=||;[KM]{q4smMdt =iKNL2{wJ + gKNLl + ^K^o]{w^ (2.13) 


where KNLl is a iunction of {Wc} only, KNL2 is a function ofboth (wc) and (wj and ICNL3 
is a function of {Ws} only These three matrices are, as well as Mb and Klb, symmetric. 


3 - STUDY OF THE STABILITY OF THE SOLUTIONS 

We will study the problem of local stability of the harmonic solution by adding 
a small disturbance to the steady state solution 

{q} = {q„} + {8qw} (3.1) 

and studying how the variation of the solution evolves. If {6q^} dies out with time 
then {q^} is stable, if it grows then {q^} is unstable. 

Inserting the disturbed solution (3.1) into equation (2.8), expanding the 
nonlinear terms into Taylor series around {q„} and ignoring terms of order higher 

than {bq^j, we obtain the variational equation: 

K]{5q„} + ^K]{6q„} + [KlJ{5q4 + ^Mj^{5q.}=M (3-2) 

The coefficients are periodic functions of time. With symbolic 

manipulation, they can easily be expanded in a Fourier series. If {q^} is of the form 

(2.10) and since is quadratic in {q„}, we have: 

3{qw} 

^(M|^=[[p.] + [p,]cos(2cst) + [p,]sin(2o)t)]. (3.3) 

2 With this formulation, KNL2 must be calculated using 2[N* J{wJ[N*J(w,} , otherwise .iKNL2 
should be considered instead of iKNL2 ■ 


1397 


Simplification (3.14) was possible because the damping matrix is, after 
transformation into modal coordinates, equal to a scalar matrix. 

Now, following Hayashi [7, page 93], we will express the solution of (3.14) in 
the form: 

|6f| = e^({bi)cos(®t) + {aj}sin(®t)) (3.15) 

which should allow us to determine, in a first approximation, the first order simple 
unstable region. 

Inserting (3.15) into (3.14) and appl 3 dng the HBM, we find 

(X^[l] + X[M,] + [M,]){y = {°} (3.16) 


0 2cd[I] 
[-2cd[I] 0 J 


[Mo] = 


where 
[M,] = 


(3.17) 


^ "1 p'" 


[BriDjB] 

[Bf[D„lB]-L^+ril]1[l]^[a>o/] 


(3.18) 


To determine the characteristic exponents, X, we transform this system into [16] 


0 [I] Ifxl 

-[M„] -[M,]JlrJ 



(3.19) 


where {X} is a vector formed by {bi}, {aj. The values of X are the eigenvalues of 
the double size matrix in the previous equation. Bearing in mind that it is the stability 
of the variable {5^} in which we are interested we substitute equation (3.15) into 
equation (3.13) to obtain 

{61} = {{b,} cos( 0 t) + {a, }sin((Dt)}. (3.20) 


If the real part of X^ - 


ii 

2 © 


is positive for any then the solution is unstable, 


otherwise it is stable. 


1398 


For undamped systems, it was demonstrated in [8], that important conclusions 
about the stability of the solutions can be deducted from the determinant of the 
Jacobian of {F}. We are going to extend the demonstration to systems with mass 
proportional damping. 

Applying the derivation rule for composite functions, we obtain the derivatives 
of {Fj} and {fJ with respect to {wc} and {wj as follows; 






Matrices [I] and [Mo] are symmetric and matrix [Mi] is skew-symmetric. This 
means that the eigenvalues of equation (3.16) are either purely imaginary or purely real 
[8]. If X is imaginary the solution is always stable; if X is real the stability limit is 
defined by 

= (3.25) 


Inserting (3.25) in (3.16) we arrive at 

■[Bf[D„][B]-co^[l] + [o.,/] [Bf[D.][B] + P l|b,l Jol 

[Bf[D„][B]-p [BnD,,][B]-<B^[l] + [co,/]JlaJ loj 

The matrix in the previous equation is [B]^[D][B], where [D] is the Jacobian 
of {F’} with respect to the vector of coefficients of the cosine and sine terms, given by 


[D] = d{F}/d 


A non-trivial solution of (3.26) exists if 

det([Bf [D][B])=0 o |BnDl=0o|Dl=0. (3.28) 

The last equivalence is true, because [B] is a non-singular matrix. Thus, we proved that 
in the stability limit, the determinant of the Jacobian of (F), |D| , is zero. 


|D| is a polynomial in the coefficients {wj and (wj and in o; therefore, it is a 

continuous function in those coefficients. All the experimental and numerical analysis 
of nonlinear vibration of beams, indicate that the shape of vibration, defined in our 
model by {wc} and {w^}, is a continuous function of the amplitude and the frequency 
of vibration. Thus, we conclude that |d| varies in a continuous way through the FRF 
(fi-equency response function) curve. Hence, if there is a change in its sign between 
two consecutive points of the FRF curve, then |D| =0 for a particular point between 

these two. In that particular point, the stability limit might have been crossed. 

So, a complete study of the first order solution’s stability may be carried out by 
determining only the characteristic exponents of the first solution and when JD] changes 
sign or when |D| is approximately zero. As jD| is needed in the continuation method and, 
when the Newton method is applied, can be easily calculated from [D], this results in 
substantial time savings. 


4-APPLICATIONS 

A clamped-clamped beam made in an aluminium alloy with the reference 7075- 
T6 was analysed. Its material [9] and geometric properties are: 

E = 7.172*10^° N/m2, p = 2800 kg/m^, h=0.002 m, b=0.02 m, L = 0.405 m 3. 

For aluminium, a typical value of the loss factor (which is multiplied by the 
stiffness matrix) is a=0.01 (C=0.5%), but the measured value in reference [18] was 
approximately equal to 0.038 (^sl.9%)^. In order to have the same damping 
coefficient for the first mode of vibration, the value of the damping factor is: 

P = (»o,^xa. (4.1) 

The beam was modelled using the HFEM, as described in section 2. To solve 
the system of equations (2.11), Newton’s method was used in the nonresonant area. In the 
vicinity of resonance frequencies it is difficult to obtain convergence by the Newton method 


3 Except in the comparison with experimental results, where L-0.406 (value of Wolfe’s clamped- 
clamped beam length). 

Wolfe did not think that the measured damping ratio was only due to material damping. He also 
attributed the obtained value to damping in the joints and to the coil magnet arrangement used to 
excite the beam. 


1400 


and a continuation method was applied [8, 14]. The derivation of the Jacobian matrix 
present in both methods was performed symbolically [15]. 

Because the excitation force will be applied in the middle of the beam and both 
the beam and the boundary conditions have symmetric properties, only symmetric out 
of-plane shape functions and antisymmetric in-plane shape functions will be used^. 

4.1 - Study of convergence with number of shape functions 

With four out-of-plane (po) and four in-plane shape functions, convergence of 
the value of the first linear natural frequency is achieved (Table 1). This number of 
shape functions will be the starting value for our nonlinear analysis. The number of 
degrees of freedom of the present damped model is equal to 2xpo. 


Table 1 - Natural linear frequencies of the cc beam (rad/s). Mode 1. 


Exact 

Po=2, pi=2 

Po=3, pi=3 

Po=4, p=4i 

Po=5, Pi=5 

396.6 

396.613 239 

396. 605 011 

396. 605 008 

396. 605 008 


In Figures 1, 2 and 3 we can see the FRFs in the vicinity of the first, third and 
fifth mode, obtained when a force P of 0.03 N was applied. Near the first mode there is 
no increase in accuracy by using more than four out-of-plane and four in-plane shape 
functions (po=4, pi=4). However, for the third mode, as the amplitude of vibration 
grows, the results obtained with po=4 and pi=4 depart from the ones obtained with 
more shape functions. The FRF curve constructed with po=5 and pi=5 is quite similar 
to the coincident FRFs obtained with po=6, pi=6 and with po=7, pi=7. In the 
neighbourhood of the fifth mode, convergence seems to be achieved with po=8 and 

pi=8. 


4.2 - Influence of in-plane displacements 

In Figure 4 the FRFs obtained considering and neglecting the in-plane 
displacements are compared. As in references [1] and [10], we found that the in-plane 
displacements ‘reduce’ the non-linearity, in the sense that the non-linearity caused by 
them is of the soft spring type and counterbalances the hard spring type non-lineanty 

5 To check if the nonlinearity introduced any coupling and consequent antisymmetric terms in the 
response, a model including symmetric and antisymmetric, in- and out-of-plane shape functions was 
considered. It was confirmed that, with these boundary conditions and with the one harmonic 
representation of the solution’s time dependency, there is no such coupling. 


1401 



caused by the transverse displacements. This ‘reduction’ of nonlinearity is due, as the 
formulation of the nonlinear stiffness matrix - eq. (2.9) - shows, to the effects of in¬ 
plane deformation on the stiffness of the structure. 

4.3 - Study of stability 

In Figures 5 and 6 we can see the stability studies carried out in the 
neighbourhood of the first and third resonance frequencies, using po=6, pi=6 shape 
functions and with an excitation force of amplitude P = 0.03 N. In all cases, |D| 
changed sign when the stability condition of the solutions changed. 

4.4 - Comparison with experimental results 

In Figures 7 and 8 we can see the comparison between the FRF obtained with 
the HFEM, using po=6, pj=6 shape functions, and the experimental results [18] when a 
force P = 0.134 N is applied in the centre of the beam. Two values were used for the 
damping factor: P=0.01 g3q, and P=0.038 g)o, . 

The HFEM provides a FRF with a slope similar to the experimental one around 
the resonance frequency. This indicates that the nonlinear stiffness is well represented 
by the model. 

The turning point corresponding to the largest amplitude of vibration, where the 
jump phenomena occurs, obtained with the HFEM, point B, does not match the 
experimental one, point A. With the typical value used for the loss factor in aluminium 
alloys, a=0.01 (p=0.01oi)oj), the maximum amplitude of vibration was more than double 
the one measured. However, the HFEM solutions represented in Figure 7 after point A are 
very close. Thus, in a real system, a small perturbation would easily make the shape of 
vibration change into an unstable one and a change, or jump, to a stable shape of vibration 
at a lower amplitude could be observed before the largest computed amplitude of vibration 
was achieved. With the measured loss factor, a=0.038, the largest amplitude of vibration 
obtained with the HFEM, was around a half of the measured maximum amplitude. 


1402 



5 - CONCLUSIONS 


The HFEM dynamic model of a beam vibrating with large amplitudes was 
constructed with small time expense. This is due to the small number of degrees of freedom with 
which convergence is achieved, to the easy way in which the number of d.o.f are reduced, 
benefiting from the symmetry properties of the problem, and to the embedding properties^ of 
the HFEM. 

For the amplitudes of vibration displacement studied, with relatively few d.o.f 
the FRF curves were accurately determined until the 5th order mode, inclusive. If 
modes of order higher than 5th are to be studied, then the inclusion of more elements 
instead of more shape functions should be considered, as shape functions of excessive 
high order turn the construction of the matrices quite time consuming. The comparison 
with experimental results showed a very good prediction of the slope of the FRF by the 
HFEM. The largest amplitude of vibration and the correspondent turning and jump 
point, are greatly influenced by the amount of damping used. 

Using the flexibility of choosing the shape functions in the HFEM model it was 
shown that the in-plane displacements cause a softening-type nonlinearity. 

With the continuation method the multi-valued regions of the FRF curves were 
completely and automatically described. 

To determine the characteristic exponents that establish the stability of the 
solution, we solved an eigenvalue problem. Due to the reduced number of degrees of 
freedom of the HFEM model this was quickly solved. More important, it was proven 
that in the stability limit the determinant |D| is zero. Thus, we only have to determine 
the characteristic exponents of the first solution and when there is an indication that 
|Dj =0 for a particular point, to check if the stability of the solution changed. This 

results in significant time savings. 

With symbolic computation, the matrices involved in the HFEM model and the 
Jacobian matrix necessary in the continuation and Newton methods, were easily and 
exactly derived, thus reducing the numerical errors. Symbolic computation was also 
helpful in the application of the HBM. 


^ Here we include the HFEM’s advantages in the derivation of the nonlinear stiffness matrix. 


1403 



REFERENCES 


[ 1 ] - Atluri, S., Nonlinear vibrations of a hinged beam including nonlinear effects. Trans, of 
the ASMS J. of Apl. Mech., 1973, 40, 121-126. 

[ 2 ] - Bermet, J. A. and Eisley, J. G., A multiple-degree-of freedom approach to nonlinear 
beam vibrations. J. of the Am. Inst, of Aeronaut, and Astronaut., 1970,8, 734-739. 

[ 3 ] - Cheung, Y. K. and Lau, S. L., Incremental time-space finite strip method for non-linear 
structural vibrations. Earthquake Engng. and Struct. Dynamics, 1982,10,239-253. 

[ 4 ] - Han, W, The Analysis of isotropic and laminated rectangular plates including 
geometrical non-linearity using the p-version finite element method^ Ph.D. Thesis, University 
of Southampton, Southampton, 1993. 

[ 5 ] - Hamdan, M.N. and Burton, T.D., On the steady state response and stability of non¬ 
linear oscillators using harmonic balance. J. of Sound and Vibr., 1993,166, 255-266. 

[ 6 ]- Stokey, W. F., Shock and Vibration Handbook, Third edition, ed. C. M. Harris, 
McGraw-Hill, New York, 1988, p. 7-14. 

[ 7 ] - Hayashi, C, Nonlinear Oscillations in Physical Systems, McGraw-Hill, New York, 
1964. 

[ 8] - Lewandowski, R., Non-linear, steady-state analysis of multispan beams by the finite 
element method. Computers and Struct., 1991,39, 83-93. 

[ 9 ] - ASM Committee on Properties of Aluminium Alloys, Properties and Selection of 
Metals, Metals Handbook, Vol. 1, 8th edition, ed. T. Lyman, Ohio, 1961, p. 948. 

[ 10 ] - Mei, C and Decha-Umphai, A finite element method for non-linear forced vibrations of 
beams. J. of Sound and Vibr., 1985,102, 369-380. 

[ 11 ] - Meirovitch, L., Elements of Vibration Analysis, McGraw-Hill, Singapore, 1986. 

[ 12 ] - Mentel, T. J., Vibrational energy dissipation at structural support junctions. In 
Colloquium on Struct. Damping, ed. E. J. Ruzicka, 1959, pp. 89-116. 

[ 13 ] - Petyt, M, Introduction to Finite Element Vibration Analysis, Cambridge University 
Press, Cambridge, 1990. 

[ 14 ] - Ribeiro, P. and Petyt, M., Study of nonlinear free vibration of beams by the 
hierarchical finite element method. ISVR Techmcal Memorandum No.773, University of 
Southampton, Southampton, November 1995. 

[ 15 ] - Redfem, Darren, The Maple Handbook, Springer-Verlag, New York, 1994. 

[ 16 ] - Takahashi, K., A method of stability analysis for non-linear vibration of beams. J. of 
Sound and Vibr., 1979, 67, 43-54. 

[ 17 ] - Wang, S. S. and Huseyn, K., Bifurcations and stability properties of nonlinear systems 
with symbolic software. Math. Comput. Modelling, 1993,18, 21-38. 

[ 18 ] - Wolfe, Howard, An experimental investigation of nonlinear behaviour of beams and 
plates excited to high levels of dynamic response, Ph.D. Thesis, University of Southampton, 
Southampton, 1995. 


1404 


FIGURES 


mnxiwi 1 

h 




0.5- 





€f^ 

().4- 






0.3- 

. g 

0.2- 

« 



O.I- 



. • 



O' 

340 360 380 4^0 420 


0 ) (rad/s) 


1 - FRF in the vicinity of the first mode of vibration. x=0.5 
0 po=4, pi=4; □ po=5, pi=5; 0 po=6, pi=6; + po=7, pi=7. 




1405 









innxlw 

h 

1 



0.5- 







9 °’ 

0.4- 



t , . 





0.3- 




0.2 







% 

0.1 





. 

. 

- * * ‘ . 

0^ 

340 

360 

380 400 420 440 




CO (rad/s) 


Figure 4 - FRF with in-plane displacements, po=6, Pi=6 (o), and without 
in-plane displacements, po=6, Pi=0 (+). x=05xL. 



Figure 5 - Stability study. First mode. x=0.5xL. □ stable solution; 
+ unstable solution; po=6, pi=6. 



1406 









mnxiwi 

h 

B 

2.2- 


2- 


1.8- 


1.6- 


1.4- 


1,2- 


I- 

A 



0.8 


0.6- 


0.4' 




0.8 1 .2 , 1.4 1.6 1.8 


CO/OD„ 


0 


Figure 7 ~ Comparison with experimental results, o HFEM stable, 

□ HFEM unstable, Po=6 and pi=6, |3=0,01cOo,2; + experimental. x=05xL. 


mnxiw 

h 

1- 



A 

0.8- 


* 


0.6 





B 



0.4 








O. 2 J 


. 




• 

* 


0.8 0.9 1 1.1 

M/Wq 

1.2 

1.3 


Figure 8 - Comparison with experimental results, o HFEM stable, 

□ HFEM unstable, po=6 and pi=6, ( 5 = 0 . 038 cOo, 2 ; + experimental. x=05xL. 


1407 








1408 


GEOMETRICALLY NONLINEAR DYNAMIC ANALYSIS 
OF 3-D BEAM 


Kuo Mo Hsiao and Wen Yi Lin 
Department of Mechanical Engineering, 
National Chiao Tung University, 
Hsinchu, Taiwan, Republic of China 


ABSTRACT 

A co-rotational finite element formulation for the geometrically 
nonlinear dynamic analysis of spatial beam with large rotations but 
small strain is presented. The deformation nodal forces and inertia 
nodal forces are derived by using the d'Alembert principle and the 
virtual work principle. The gyroscopic effect is considered here. 

The beam element developed here has two nodes with sbc degrees 
of freedom per node. Some angular velocity coupling terms, which 
are so called gyroscopic forces, are obtained in inertia nodal force. 

An incremental-iterative method based on the Newmark direct 
integration method and the Newton-Raphson method is employed 
here for the solution of the nonlinear dynamic equilibrium 
equations. Numerical examples are presented to demonstrate the 
acctuacy and efficiency of the proposed method. 

INTRODUCTION 

In recent years, the nonlinear dynamic behavior of beam 
structures, e.g., framed structures, flexible mechanisms, and robot 
aims, has been the subject of considerable research. In [1], Hsiao ^d 
Jang presented a co-rotational formulation and numerical 
procedure for the dynamic analysis of planar beam structures. This 
formulation and numerical procedure were proven to be very 
effective by numerical examples studied in [1]. However, it is only 
limited for planai* beam structures. A general formulation for three 
dimensional beam element is not a simple extension of a two 
dimensional formulation, because large rotations in three 
dimensional analysis are not vector quantities; that is, they do not 
comply with the rules of vector operations. In [2] a motion process 
of the three dimensional beam element is proposed for the large 
displacement and rotation analysis of spatial frames. In [3] a co- 
rotational formulation for three-dimensional beam element is 
proposed. However, it is only limited for nonlineai' static analysis. 

The objective of this study is to present a practical formulation for 
the dynamic analysis of three dimensional Euler beam. The 
kinematics of the beam element proposed in [3] is adopted here. 


1409 



The element deformations are determined by the rotation of 
element cross section coordinates, which are rigidly tied to element 
CTOSS section, relative to the element coordinate system [2, 3]. The 
three rotation parameters proposed in [3] are used to determine the 
orientation of the element cross section coordinates. In order to 
capture the gyroscopic effect, the relation between the time 
derivatives of the rotation parameters and the angular velocity and 
the angular acceleration is derived here. The beam element 
developed here has two nodes with six degrees of freedom per node. 
The element nodal forces are conventional forces and moment. 
The deformation nodal forces and inertia nodal forces are derived 
by using the d'Alembert principle and the virtual work principle in 
the current element coordinates. An incremental-iterative method 
based on the Newmark direct integration method and the Newton- 
Raphson method is employed here for the solution of the 
nonlinear dynamic equilibrium equations. Numerical examples 
are presented to demonstrate effectiveness of the proposed method. 

FINITE ELEMENT FORMULATION 
Basic assumptions 

The following assumptions are made in the derivation of the 
nonlinear behavior: (1) the beam is prismatic and slender, and the 
Euler-Bernoulli hypothesis is valid; (2) the centroid and the shear 
center of the ctoss section coincide; (3) the unit extension and twist 
rate of the centroid axis of the beam element are uniform; (4) the 
cross section of the beam element does not deform in its own plane, 
and strains within this cross section can be neglected; (5) the out-of¬ 
plane warping of the cross section is the product of the twist rate of 
the beam element and the Saint Venant warping function for a 
prismatic beam of the same cross section; (6) the deformations of the 
beam element are small. 

Coordinate systems 

In this paper, a co-rotational total Lagrangian formulation is 
adopted. In order to describe the system, following [3], we define 
three sets of coordinate systems (see Fig. 1): 

(1) A fixed global set of coordinates, Xj(z = 1,2,3); the nodal 
coordinates, displacements, and rotations, and the stiffness matrix 
of the system are defined in this coordinates. 

(2) Element cross section coordinates, rf (f = 1,2,3); a set of element 
cross section coordiaates is associated with each cross section of the 
beam element. The origin of this coordinate system is rigidly tied to 


1410 


the shear center of the cross section. The xf axes are chosen to 
coincide with the normal of the corresponding cross section and the 

X 2 and X 3 axes are chosen to be the principal directions of the cross 
section. 

(3) Element coordinates, x, (z = 1,2,3); a set of element coordinates 
associated with each element. The origin of this coordinate system 
is located at node 1 ; the xj axis is chosen to pass through two end 
nodes of the element, and the X 2 and X 2 axes are determined from 
the orientation of the element cross section coordinates at two end 
nodes using the way given in [2]. The deformations and stiffness 
matrices of the elements are defined in terms of this coordinates. 
In this paper the element deformations are determined by the 
rotation of element cross section coordinates relative to this 
coordinate system. 

Rotation vector and rotation parameters 
For convenience of the later discussion, the term 'rotation vector' 
is used to represent a finite rotation. Figure 2 shows that a vector 
b which as a result of the application of a rotation vector (pa. is 
transported to the new position h'. The relation between b and b' 
may be expressed as [4] 

b' = cos 0 b + (1 - cos 0 )(a • b) + sin 0 (a x b), ( 1 ) 

where (p is the angle of counterclockwise rotation, and a is the unit 
vector along tiie axis of rotation. 

In this paper, the s)mbol { } denotes column matrix. Let e,- and ef 
(i - 1 , 2 , 3 ) denote the unit vectors associated with the x, and xf axes, 

respectively. Here, the traid ef in the deformed state is assumed to 
be achieved by the application of the following two rotation vectors 
to the traid Cj : 

e„ = 0 „n, = 0 it, 

where 

n={0, 82! (el + 03/(0! + 0|)V2} 

= {0,^2,713}, 
t = {cos 0 „, 62 , 63 }. 

cos 0 „=(i- 0 !-e!)Vl 

^ dw{s) . dv(s) 


(2,3) 


( 4 ) 


1411 


in which n is the unit vector perpendicular to the vectors ei and 
ef, and t is the tangent unit vector of the deformed centroid axis. 

Note that ef coincides with t. is the inverse of cosd„. v(s) and 
w(s) are the lateral deflections of the centroid axis of the beam 
element in the X 2 and directions, respectively, and s is the arc 
length of the deformed centroid axis. 

The rotation vectors e„ and 0^ are determined by (f = l,2,3). 
Thus, di are called rotation parameters in this study. 

Using Eqs. (l)-(4), the relation between the vectors and ef 
(i = 1,2,3) in the element coordinate system may be obtained as 

ef = [t, Ri, R2] = Re,-, 

Rl = cos^iri + sin0ir2, 

R 2 = -sin^iri + cos0ir2, 

ri = {-03, cos 0„ + (1 - cos 0„, (1 - cos d„)n 2 n^h 
r2 ={02^(1-cos0„)w2W3,cos 0„ +(l-cos0„)n3}, (5) 


where R is the so-called rotation matrix. 

Let 0 = {01, 02, 03> be the vector of rotation parameters, 36 be the 

variation of 0. The traid ef corresponding to 0 may be rotated by a 
rotation vector = {3ipi, 3(l>2, <5^3} to reach their new positions 
corresponding to 0 + 50 [3]. When 02 and 03 are much smaller 
than unity, the relationship between 50 and 5<^ may be 
approximated by 


r 1 


50 = 


-03 


^2 


03/2 -02/2' 
1 0 

0 1 




( 6 ) 


If both sides of Eq. (6) is divided by St, the first time derivative of 0 
may be expressed by 


r 1 


0 = 


-03 


02 


03/2 - 02 / 2 - 

1 0 

0 1 


(j) = T^^ij), 


(7) 


where the symbol ( ) denotes differentiation with respect to time t . 


1412 


= 1,2,3) denote the angular velocities about the axes. 

From Eq. (7), the second time derivative of 6 may be expressed by 
may be expressed as 

9 = t-^(j) + T"^^, (8) 


where = 1,2,3) denote the angular accelerations about the Xi 
axes. 

Nodal parameters and forces 

The global nodal parameters for the system of equations 
corresponding to the element local nodes j (j - 1, 2) are ll^j, the Xj 
(i = 1,2,3) components of the translation vectors at nodes j, and 
the Xi (f = 1,2,3) components of the rotation vectors at 
nodes j. Here, the values of are reset to zero at current 
configuration. Thus, <50^, the variations of ^ij, represent 
infinitesimal rotations about the Xi axes [3], <i>ij and Oy represent 
angular velocities and angular accelerations about the Xj axes, 
respectively. Tlie generalized nodal forces corresponding to dOy are 
the conventional moments about the Xf axes. The generalized 
nodal forces corresponding to dllij, the variation of Uy-, are the 
forces in the X, directions. 

The element employed here has six degrees of freedom per node. 
Two sets of element nodal parameters termed 'explicit nodal 
parameters' and 'implicit nodal parameters' are employed. The 
explicit nodal parameters of the element are used for the assembly 
of the system equations from the element equations. Thus, they 
should be consistent with the global nodal parameters, and are 
chosen to be Uij, the x, (i = l,2,3) components, of the translation 

vectors uj at nodes j (j = 1, 2) and 0y, the Xi (i = 1,2,3) components 
of the rotation vectors (j)j at nodes j. Similarly, the generalized 
nodal forces corresponding to Wy and d^ij are /y and my, the forces 
in the Xi directions and the conventional moments about the Xj 
axes, respectively. 

The implicit nodal parameters of the element are used to 
determine the deformation of the beam element. They are chosen 
to be My, the Xj (i = l,2,3) components of the translation vectors u^ 
at nodes j and 0y, the nodal values of the rotation parameters 0- 


1413 


(i = 1,2,3) at nodes ; (j = 1, 2). The generalized nodal forces 
cori'esponding to duij and dOij are and m|, the forces in the Xi 
directions and the generalized moments, respectively. Note that 
are not conventional moments, because S 6 ij are not 

infinitesimal rotations about the Xi axes. 

In view of Eq. (6), the relations between the variation of the 
implicit and explicit nodal parameters may be expressed as 




dui' 


1 

0 

0 

0 

dui 

dQi 


0 

TI^ 

0 

0 

5<i>i 

' dU2 

► — 

0 

0 

I 

0 

5u2 

502 


0 

0 

0 

Ti\ 

^^2 




(9) 


where daj = {duij,du 2 j,Su 3 j}, d 6 j={ddij, 5 d 2 j, 862 j}, and 
% ={ 8 <hj’^<p 2 j>^<p 3 jh (j = % 2). I and 0 are the identity and zero 

matrices of order 3x3, respectively. (j = 1 , 2) are nodal values 

of T"l 

Let f = {fi,mi,f2,m2K f® ={fi,mf,f2,m^}, where ij ={fij,f2jj3jh 
and mj (/’ = h 2), denote the 

internal nodal force vectors corresponding to the variation of the 
explicit and implicit nodal parameters, dq and, <5q^, respectively. 
Using the contragradient law [5] and Eq. (7), the relation between f 

and, f ^ may be given by 

f = (10) 


Kinematics of beam element 

The deformations of the beam element are described in the current 
element coordinate system. From the kinematic assumptions made 
in this paper, the deformations of the beam element may be 
determined by the displacements of the centroid axis of the beam 
element, orientation of the cross section (element cross section 
coordinates), and the out-of-plane warping of the cross section [3]. 
Let Q (Fig. 1) be an arbitrary point in the beam element, and P be the 
point corresponding to Q on the centroid axis. The position vector 
of point Q undeformed and deformed configurations may be 

expressed as 


1414 


and 


r^xei + ije2 + ze^, 


( 11 ) 


r = + ^(s)e2 + ^(s)e3 + yel + ze| + Si^s^Qv (1^) 

where Xf,{s), v{s) and w{s) are the xy X 2 and X't, coordinates of point 
P, respectively, s is the arc length of the deformed centroid axis 
measured from node 1 to point P. ^c(®) expressed by 

Xc{s) = Mil + 

where un is the displacement of node 1 in the xi direction, and 
cos0„ is defined in Eq. (4). 

Here, z;(s) and w{s) in Eq. (12) are assumed to be the Hermitian 
polynomials of s , and 0i(s) in Eq. (12) is assumed to be linear 
polynomials of s, and maybe given by 

z;(s) = {Ni,N2 ,N3,N4}^{W21/%1/W22/%2}-= 

w{s) = {N'i,-N2,N3-,N4}^{m31,021/^32^^22} = 

0l(s) = {N5,N6}^{0n^%> = N^u^, (14) 

Ni = 7(1 - ?)^(2 f I), Nj = |(1 -1^)(1 -1). 

4 o 

yvj = i(i +1)2(2- f), Ni = |(-1 + ?2)(1 +1). 

4 o 

Af5 = |(l-?). W6=|(l + |), (15) 

where S is the arc length of the centroid axis of the beam element 
and may be expressed by 

S^ltjQOS (17) 

where I is the chord length of the centroid axis of the beam 
element, and cos is given in Eq. (4). 

The way to determine the current element cross section 


1415 


coordinates at both ends, element coordinates, and element implicit 
nodal parameters corresponding to displacement increments is 
given in [ 2 ,3]. 

If ;c, y, and z in Eq, (11) are regarded as the Lagrangian 
coordinates, the Green strains en, Si 2 , and 613 are given by 


£11 = 


1 

2 


\dxj [dxj 


-1 


£12 


2[^xJ 


2[^xJ \dzj' 


(18) 


Substituting Eqs. (4), (5), (12), and (13) into Eq. (18), sn, ei 2 , and £13 
can be calculated. 

Element nodal force vector 

The element nodal force vector (Eq. (10)) corresponding to the 
implicit nodal parameters are obtained from the d'Alembert 
principle and the virtual work principle. For convenience, the 
implicit nodal parameters are divided into four generalized nodal 
displacement vectors u,- (i = a,b,c,d), where 

Urt «{«!!, «i2}/ 

and u^, Uc, and u^ are defined in Eq. (14).. 

The generalized force vectors corresponding to (5Uj, the variation 
of u,- (f = a,h,c,d) are 


4 - + 4 = 


where f^ and f*- {i = a,b,c,d) are the deformation nodal force vector 
and the inertia nodal force vector, respectively. 

The virtual work principle requires that 


1416 


du^Ja + < 50^4 + dn^ic + 

=JL {oiideii + 2 ai 2 dei 2 + 2 ai 3 ( 5 ei 3 + p8r^r)dV, 

where on = Esiy O 12 = 2Gei2 and 0-13 = 2Gei2, where E is tl^ 
Young's modulus and G is shear modulus, p is the density, and V 
is the volume of the undeformed beam. 

If the element size is properly chosen, the values of the nodal 
parameters (displacements and rotations) of the element defined in 
the current element coordinate system, which are the total 
deformational displacements and rotations, may always be much 
smaller than unity. Thus only the first order terms of nodal 
parameters are retained in deformation nodal forces. However, in 
order to include the effect of axial force on the lateral forces, a 
second order term of nodal parameters is retained. Because the 
values of the nodal parameters of the element may always be much 
smaller than unity, it is reasonable to assume that the coupling 
between the nodal parameters and their time derivatives are 
negligible. Thus only zeroth order terms of nodal parameters are 
retained in inertia nodal forces. 

From Eqs. (4), (5), and (12)-(21), the deformation nodal forces and 
the inertia nodal forces may be expressed as 


Li 

(22,23) 

. . rd Gj{di2-Qn)f 111 

f“ = + k^)Uc, td- ^ i 

(24, 25) 

fa = 

(26) 

4 = + m^)Ub - 2p4 , 

(27) 

fc = - 2ply fN'cdidsds , 

(28) 

f^ = m^Urf - p{ly - 

(29) 


1417 


where A is title cross section area, L is the initial length of the beam 
element, k,- and k^- {i = b,c) are bending and geometric stiffness 
matrices of conventional beam element [5,6], and J is the torsional 
constant p is the density, ly and 7^ are the moment of inertia of 

the beam cross section about the and axes respectively, m^ is 
the consistent mass matrix of bar element for axial translation, nij-f 
and m,> {i^b,c) are the consistent mass matrices of elementary 
beam element for lateral translation and rotation, respectively, and 
is the consistent mass matrix of bar element for axial rotation. 
These mass matrices can be found in [5, 6]. The underlined terms in 
Eqs. (27)-(29) are inertia forces induced by the gyroscopic effect, and 
are called gyroscopic forces. 

Element Matrices 

The element stiffness matrices and mass matrices may be obtained 
by differentiating the element nodal force vectors with respect to 
nodal parameters, and time derivatives of nodal parameters. 
However, element matrices are used only to obtain predictors and 
correctors for incremental solutions of nonlinear equations in this 
study. Thus, approximate element matrices can meet these 
requirements. The stiffness matrices and mass matrices of 
elementary beam element given in [5, 6] are also used here. 

Equations of motion 

The nonlinear equations of motion may be expressed by 

F^ = F^ + F^-P = 0 (30) 

where F^ is the unbalanced force among the inertia nodal force F^, 
deformation nodal force F^, and the external nodal force P. F^ and 

F^ are assembled from the element nodal force vectors in Eq. (10), 
which are calculated using Eqs. (10) and (22)-(29) first in the current 
element coordinate system, and then transformed from current 
element coordinate system to global coordinate system before 
assemblage using standard procedure. 

APPLICATIONS 

An incremental iterative method based on the Newmark direct 
integration method and the Newton-Raphson method is employed 
here for the solution of the nonlinear d 5 mamic equilibrium 
equations. 


1418 


The example considered is a right-angle cantilever beam subjected 
to an out-of-plane concentrated load as shown in Fig. 3. Four 
elements are used for discretization. A time step size of At = 0.25^ is 
used. The cantilever undergoes a finite free vibration with 
combined bending and torsion after the removal of the applied 
load; the time histories of out-of-plane displacements of the elbow 
and of the tip are given in Figs. 4 and 5. It is seen that the present 
results are in excellent agreement with those given in [7] and [8]. 
However, it should be mentioned that the beam elements used in 

[7] and [8] are derived using fully nonlinear beam theory and total 
Lagrangian formulation. Thus, the beam elements used in [7] and 

[8] are much more complicated than that proposed here. 

CONCLUSIONS 

A co-rotationai finite element formulation for the geometrically 
nonlinear dynamic analysis of spatial beam with large rotatio^ but 
small strain is presented. The deformation nodal forces and inertia 
nodal forces are derived by using the d'Alembert principle and the 
virtual work principle. The gyroscopic effect are considered here. 

The nodal coordinates, displacements, rotations, velocities, 
accelerations, and the equation of motion of the system are defined 
in a fixed global set of coordinates. The beam element developed 
here has two nodes with six degi'ees of fi^eedom per node. The 
element nodal forces are conventional forces and moments. AU of 
element deformations and element equations are defined in terms 
of element coordinates which are constructed at the current 
configuration of the beam element. The element deformations are 
determined by the rotation of element cross section coordinates, 
which are rigidly tied to element cross section, relative to the 
element coordinate system. In conjunction with the co-rotational 
formulation, the higher order terms of nodal parameters in 
element nodal forces are consistently neglected. 

An incremental-iterative method based on the Newmark direct 
integration method and the Newton-Raphson method is employed 
here for the solution of the nonlinear dynamic equilibrium 
equations. Numerical examples are presented to demonstrate the 
accuracy and efficiency of the proposed method 

It is believed that the co-rotational formulation for 3-D beam 
element presented here may represent a valuable engineering tool 
for the d 5 mamic analysis of spatial beam structures. 

ACKNOWLEDGMENT 

The research was sponsored by the National Science Council, 
Republic of China, under contract NSC86-2212-E-009-006. 


1419 


REFERENCES 

1. Hsiao, KM. and Jang, J.Y., . Nonlinear dynamic analysis of elastic 
frames. Computers & Structures, 1991, 33, 769-781. 

2. Hsiao, KM. and Tsay, C.M., A motion process for large 
displacement analysis of spatial frames. International Journal of 
Space Structures, 1991, 6, 133-139. 

3. Hsiao, K.M., Corotational total Lagrangian formulation for 
three-dimensional beam element. AIAA Journal, 1992, 30, 797- 
804. 

4. Goldstein, H., Classical Mechanics, Addision-Wesley, Reading, 
MA,1980. 

5. Dawe, D.J., Matrix and Finite Element Displacement Analysis of 
Structures, Oxford Univ. Press, New York, 1984. 

6. Hsiao, I<.M., A study on the dynamic response of spatial beam 
structures. NSC 82-0401-E009-081 Report, National Science 
Council, Taiwan, Republic of China,, 1993. 

7. Simo, J.C. and Vu-Quoc, L., On the dynamics in space of rods 
undergoing large motions - A geometrical exact approach. 
Computer Methods in Applied Mechanics and Engineering, 
1988, 66,125-161. 

8. lura, M. and Atluri, S.N., Dynamic analysis of finitely stretched 
and rotated three-dimensional space-curved beams. Computers 
& Structures, 1988, 29, 875-889. 



Fig. 1 Coordinate systems. 


1420 


Fig.2 Rotational vector 




Material Properties: 
EA==10" 

EIy = EI^=GJ = 10^ 

Ap = 1 

ply=pl,= 10 


Time History of Load: 



Fig.3 Right-angle Cantilever beam 



0 5 10 15 20 25 30 


Time 


Fig. 4 Displacements in the X 3 direction at point B. 



Fig. 5 Displacements in the X 3 direction at point A. 


1422 









Nonlinear Response of Composite Plates to 
Harmonic Excitation Using The Finite Element 
Time Domain Modal Method 


Raymond Y, Y. Lee, Yucheng Shi and Chuh Mei 
Department of Aerospace Engineering 
Old Dominion University, Norfolk, VA 23529-0247 

Abstract 

A multimode time domain formulation based on the finite element method for large 
amplitude vibrations of thin composite plates subjected to a combined harmonic excitation 
and thermal load is presented. By using the modal reduction method, the system equations of 
motion in physical coordinates are transformed into the linear modal coordinates and the 
sizes of the system matrices are reduced drastically. The reduced system modal equations can 
be handled easily with less computational efforts. The jBrequency-maximum deflection 
relations of simple harmonic, superharmonic and subharmonic responses are predicted by 
choosing suitable initial conditions. The procedure for the selection of the initial conditions is 
also presented. A laminated composite plate is studied in great detail. External loadings 
considered are harmonic excitations or combined harmonic and thermal loads. The steady 
state responses of the linear modal coordinates are presented in details at several frequencies. 
Their phase plots, power spectrums and time domain graphs are given and discussed. 


Introduction 

The increase use of advanced composites as high performance structural 
components necessitates accurate prediction methods which reflect their 
multilayered anisotropic behavior. Thin laminated composite plates subjected to 
severe harmonic lateral loadings are likely to encounter flexural oscillations 
having amplitudes of the order a plate thickness. For the prediction of forced 
vibration response, the multilayered anisotropic behavior, the complex 
boundary conditions, and the complex loading cases such as the present of the 
thermal loads make the problem even more difficult. Methods of analysis 
dealing with large deflections are thus becoming increasingly important. 

Whitney and Leissa [1] have formulated the basic governing equations for 
nonlinear vibrations of heterogeneous anisotropic plates in the sense of von 
Karman. Based on those equations, a number of classical continuum 


1423 


approaches exists for the analysis of nonlinear plate behavior. In general, the 
Galerkin’s method is used in the spatial domain, where the plate deflection is 
expressed in terms of one or more linear vibrational mode shapes; and various 
techniques in the temporal domain such as the direct numerical integration, 
harmonic balance, incremental harmonic balance, perturbation, and multiple 
scales methods, to cite a few, are employed. Excellent collections of classical 
continuum solutions and reviews on geometrically nonlinear analysis of 
laminated composite elastic plates are given by Chia [2,3] and Sathyamoothy 
[4]. The internal resonance of nonlinear systems has been thoroughly 
investigated using the multiple scales by Nayfeh and Mook [5]. Most recently, 
Wolfe et al. [6] have reviewed various analytical methods and have obtained 
experiment data on beams and plates excited sinusoidally or randomly. Most of 
the classical continuum solutions of composite plates have been limited to 
single-mode approximation. This is due to the difficulties in obtaining the 
general Duffing-type multiple-mode equations using the Galerkin’s approach 
especially for arbitrarily (unsymmetrically) laminated composite plates with 
complex boundary conditions. 

The finite element method has proven to be a powerful and versatile 
approach for structural problems of complex geometries, boundary conditions, 
and loadings. Reddy [7] has reviewed the application of finite element methods 
to linear and nonlinear anisotropic composite plate problems. In this paper, the 
nonlinear steady state periodic responses of thin rectangular arbitrarily 
laminated composite plates excited sinusoidally with or without the presence of 
thermal load are presented using the finite element time domain modal method. 
A rectangular composite plate is studied in detail. 


Formulation 

The finite element system equations of motion for large amplitude vibrations 
of a thin laminated composite plate can be expressed as 

[M]{w}+[C]{w}+([K]-[Knt]+[K 1(W)]+[K2(W)]){W} = {P(t)}+{PT} 

( 1 ) 

where [M], [C], [K], [Knt] , {P(t)} and {Pt} are constant matrices and vectors 
and represent the system mass, damping, linear stiffness, thermal effort and 


1424 


loads, respectively; and [Kl] and [K2] are the first and second order nonlinear 
stiffness matrices and depend linearly and quadratically on the unknown 
structural nodal displacements {W}, respectively. The derivation of the element 
matrices and load vectors and their explicit expressions are referred to 
references [8,9]. 

The system equations of motion presented in eq. (1) are not suitable for 
direct numerical integration because: a) the nonlinear stiffness matrices [Kl] 
and [K2] are functions of the unknown nodal displacements, and (b) the 
number of degrees of freedom (DOF) of the system nodal displacements {W} 
is usually too large. Therefore, eq. (1) has to be transformed into the modal or 
generalized coordinates of much smaller DOF. Various reduction methods for 
nonhnear problems have been summarized in an excellent review article by 
Noor [10]. For nonlinear dynamic problems, the base vectors need updating 
using the modal methods presented in [10]. In the present formulation, the 
forced general Duffing-type modal equations will have constant nonlinear 
modal stiffness matrices, therefore updating of the base vectors is not needed. 
This is accomplished by a modal transformation and truncation 


{W} = ^q,(t){(|)}">=[<I>]{q} (2) 

r=l 

where the system mode shapes are the solution from the linear eigen-problem 
cOf [M]{(j)}^^^ =[K]{({)}^^\ The nonlinear stiffness matrices [Kl] and [K2] in 
eq. (1) can now be expressed as the sum of the products of modal coordinates 
and nonlinear modal stiffness matrices as 

[Kl] = ^qr[Kl«)W)] (3) 

r=l 

and 

[K2] = ^ ^q^qs [K2((j)<">(t)<^’)](4) 

r=l s=l 


The nonlinear modal stiffness matrices [Kl]^"^ and [KZ]^"®^ are assembled from 
the element nonlinear modal stiffness [kl]^''^ and [k2]^^^^ as 


1425 



(5) 


([K1]">,[K2]'”>) = ^([kl]® ,[k2]<“>) 

all elements 
+ bdy. conds. 

where the element nonlinear modal stiffness matrices are evaluated with known 
system linear mode W. Thus, the nonlinear modal stiffness matrices 
and are constant matrices. Equation (1) is thus transformed to the 

forced general Duffing-type modal equations as 

[M]{q}+[c]{q}+([K]+[Klq]+[K2qq]){q} = {F(t)} (6) 


where the modal mass, damping, and linear stiffness matrices are 

([m], \c\ [k]) = [OJ'T ([M], [C], [K] - [Kot ])[<I.] (7) 


and the quadratic and cubic terms in modal coordinates and the modal force 
vector are 


[Klq]{q} = [<E.f 2^qr[Kl]« 

lr=l 

[.K2qq]{q} = [3.]^ ^^q,q,[K2] 


[<E>]{q} 


Vr=l s=l 


( 8 ) 

(9) 


{F} = [cI>f({P(t)}+{PT}) (10) 

AH modal matrices in eq. (6) are constant matrices. With given initial 
conditions, the response of modal coordinates {q} can be determined from eq. 
(6) with any direct numerical integration scheme such as the Runge-Kutta or 
Newmark-P method. Therefore, no updating of the vibration modes is needed. 
The following is the description of the selection of the initial conditions for 
periodic motions. 

With the input of suitable initial conditions, three types of solutions, periodic 
or nearly simple harmonic, superharmonic and subharmonic solutions, can be 
obtained. The selection of each type solution is based on the solution of the 1- 


1426 


DOF Duffing equation obtained by using the modal reduction method 
described earlier. For example, the system equations of motion of a symmetric 
composite plate can be reduced to 1-DOF model (Note: the quadratic term is 
gone because the plate is symmetric) as 

Mjq,+Crqr+Krqr+K2,q^ =FrSin(C0t) (11) 

where Mr, Cr, Kr, K 2 r and Fr are scalar constants and represent the modal mass, 
damping, stiffness and force; co is the forcing frequency and the subscript “r” 
denotes the linear modal number. The solution of eq. (11) can be assumed as 
=AiCos(cot)+ A 3 cos( 3 cot) for the simple harmonic and superharmonic 
solutions, then two sets of Ai and A 3 can be obtained by the substituting of the 
assumed qr into eq.(n). One set is for the simple harmonic solution and the 
other set is for the superharmonic solution. Based on these solutions, the initial 
displacement of the r-th modal coordinate in eq. (6) is chosen as A 1 +A 3 . The 
initial velocities and all other initial displacements are zero. Similarly, it is 
assumed that qr =Aicos(cot)+ Ai/3Cos(cot/3) for the subharmonic solution. 
Then, all those initial conditions can be found by repeating the procedure just 
described. 


Results and Discussions 

A simply supported eight-layer symmetrically laminated (0/45/-45/90)s 
composite plate is studied in great details. The plate isofl5xl2x 0.048 in. 
(38.1 X 30.5 X 0.122 cm). The inplane boundary conditions are immovable, i.e. 
u=v=0 on all four edges. The graphite-epoxy material properties are : Ei=22.5 
Msi (155 GPa), E 2 =1.17 (8.07), G 12 = 0.66 (4.55), V 12 = 0.22 and p = 0.1458 
X 10'^ ib-sVin^ (1550 kg/m^). The C‘ conforming rectangular plate element is 
used in the finite element model and the plate is modeled with 12 x 12 (144 
elements) mesh. The element has a total of 24 DOF (16 bending and 8 
membrane). The lowest six natural frequencies (cOr, r=l,6) and their 
corresponding mode shapes are : coi = 55.46 Hz for (1,1) mode, Oh = 125.736 
Hz for (2,1) mode, CO 3 = 151.951 Hz for (1,2) mode, CD 4 = 216.475 Hz for (2,2) 
mode, CO5 =250.585 Hz for (1,3) mode and cOe = 310.774 Hz for (3,1) mode. 


1427 


Two load cases considered are uniformly distributed harmonic excitation over 
the plate with and without the piresence of temperature. A constant modal 
damping factor, = C,/(2Mr 0)r), of 0.02 and a four-mode solution are used in 
the examples (Only the (1,1), (2,2), (1,3) and (3,1) modes are considered 
because the uniformly distributed excitation cannot induce any response of the 
(1,2) and (2,1) modes, see Table 7 of [11]). 

Harmonic Excitations 

A uniformly distributed pressure load of the form p(x,y,t) =po sincot is 
considered for the forced vibration problem. The force intensity is maintained 
at po = 0.00438 psi (30.2 Pa), however, the forcing frequency co is varying in a 
wide range from 0 to 4.5 times of the lowest linear natural frequency ©i. The 
results are shown in Figs. 1 to 4, where the designations for the total responses 
are indicated in Fig. 1, while the time-histories, phase plots and power spectra 
are given in Figs. 2 to 4. To make clear the behavior of the vibration response 
at particular frequency, each modal coordinate is depicted for understanding 
the simple harmonic, superharmonic and subharmonic response of the nonlinear 
system. 

Figures 2a-c correspond to the responses of the three forcing frequencies at 
0.6, 2.4 and 3.8 times of ©i labeled as (1) to (3) in Fig. 1. It can be seen that 
the total response of the centre of the plate is dominated by the first mode (It 
should be noted that the centre of the plate has zero contribution from the (2,2) 
mode). The frequency responses of the four modal coordinates are composed 
of superharmonic frequency components of order 2, 3, 5 .. etc. , as well as the 
input driving force frequency. At the frequency of the point (3) in Figs. 2c and 
2 d, the corresponding time histories of the second, third and fourth modal 
coordinates are unsymmetric as that of the first modal coordinate is symmetric. 
Hence, the plate is vibrating with a non-zero equilibrium position due to the 
unsymmetric responses of the second, third and fourth modes. 

In Figs. 3a and 3b, which correspond to the responses of two points (4) and 
(5) at © = 2.4 ©1 and 4.2 ©i in Fig. 1, the total response of the centre of the 
plate is almost pure simple harmonic at that particular frequency range. 

In Fig. 4a, which corresponds to the response of the point (6) at © = 3.8 ©i 
in Fig 1, the centre of the plate is mainly composed of subharmonic response of 
order 1/3. In the time histories of the four modal coordinates of Fig. 4b, it can 
be seen that the subharmonic component in the total response is contributed by 


1428 


the (1,1) modal coordinate, and the responses of the higher modal coordinates 
are pesudo harmonic. 

Combined Harmonic and Thermal Loads 

In addition to the uniform pressure po sincot, a steady state temperature 
change of 2.9Tcr is also applied to the composite plate ( where the buckling 
temperature =13.79 °F). The forcing frequency is taken as co = 1.45c0i and 
it is kept at that constant frequency of excitation, however, three pressure 
intensities at po = 6, 10 and 14 x 10*^ psi are considered. The responses are 
shown in Figs. 5a-c. The results are shown after the transient response being 
damped out, this is demonstrated by the quasi-steady state time histories in 
Figs. 5a and 5c. 

When the pressure load is small at 0.006 psi (41.3 Pa), the plate exhibits 
small oscillations about one of the thermally buckled positions (Wmax/h 
=1.0237) shown in Fig. 5a. With increase of the pressure loading, the amplitude 
of vibration increases. Fig. 5b shows the so-called snap-through or oil-canning 
phenomenon at po = 0.010 psi (68.9 Pa), the plate behavior is chaotic and has 
two potential weUs. With the further increase of the pressure loading, the plate 
exhibits large amplitude oscillations through the two buckled positions as 
shown in Fig. 5c at po = 0.014 psi (96.4 Pa). The plate motion is periodic at 
low and high pressure loads, however, the plate response composes of 
superharmonic frequency components of order 2 and 3 at the low pressure and 
of order 3 and 5 at the high pressure as shown in the PSD plots. 

The substance of the transition of the three distinct plate behaviors, from the 
small oscillations into the chaotic motion then into the large amplitude 
vibrations with the increase of forcing intensity, is shown in Fig. 6. In the low 
pressure range, the plate could also vibrate about the another equally possible 
bifurcation buckled position shown with dotted lines. 


Conclusion 

Based on the finite element method, a multimode time-domain formulation 
for nonlinear forced vibration of composite plates is presented. The main 
advantage of this method is that the system matrix equation derived from the 
FEM can be transformed into a set of general type Duffing equations with 
constant system matrices and much smaller DOF. The selections of initial 


1429 


conditions for the subharmonic, simple harmonic and superharmonic responses 
are presented. Through detailed descriptions, the frequency response 
characteristics, phase plots, time histories and the power spectrums have been 
illustrated for the three types of responses. The responses of a thermally 
buckled composite plate under harmonic excitation with fixed forcing 
frequency and various amplitudes are also obtained. Snap-through motion is 
observed at moderate pressure loads. 


References 

1. J. M. Whitney and A. W. Leissa 1969 Journal of Applied Mechanics 36, 261-266. 
Analysis of heterogeneous anisotropic plates. 

2. C. Y. Chia 1988 Applied Mechanics Review 41, 439-451. Geometrically nonlinear 
behavior of composite plate: A review. 

3. C. Y. Chia 1980 Nonlinear Analysis of Plates, McGraw-Hill, New York. 

4. M. Sathyamoorthy 1987 Applied Mechanics Review 40, 1553-1561. Nonlinear 
vibration analysis of plates: A review and survey of current developments. 

5. A. H. Nayfeh and D. T. Mook 1979 Nonlinear Oscillations, John Wiley, New 
York. 

6. H. F. Wolfe, C. A. Shroyer, D. L. Brown and L. W. Simmons 1995 Technical 
Report WL-TR-96-3057, Wright Laboratory, Wright Patterson AFB, Ohio. An 
experimental investigation of nonlinear behavior of beams and plates excited to 
high levels of dynamic response. 

7. J. N. Reddy 1985 Shock and Vibration Digest 17, 3-8. A review of the literature 
on finite element modeling of laminated composite plates. 

8. Y. Shi and C. Mei 1996 Proceedings of the 37th AIAA / ASME / ASCE / AHS / 
ASC Structures, Structural Dynamics, and Material Conference, Salt Lake City, 
UT, 1355-1362. Coexisting thermal postbuckling of composite plates with initial 
imperfections using finite element modal method. 

9. C. K. Chiang, C. Mei and C. E. Gray, Jr. 1991 Journal of Vibration and Acoustics 
113, 309-315. Finite element large-amplitude free and forced vibrations of 
rectangular thin composite plates. 

10. A K. Noor 1981 Composites and Structures, 13, 31-44. Recent advances in 
reduction methods for nonlinear problems. 

11. Y. Y. Lee, Y. Shi and C. Mei 1997 Proceedings of the 6th International 
Conference on Recent Advances in Structural Dynamics, University of 
Southampton, UK. A finite element time domain multi-mode method for large 
amplitude free vibration of composite plates. 


1430 


Centre Disp/Thlckness Centre Disp/Thickness 


Po=0.00438 Psi 
Damp. Ratio = 0.02 


(3) 


( 2 ) 


( 1 ) 


■ ■ 


(4) 


(6) 


(5) 


0 1 2 oa/osl 3 4 5 

Freq. Ratio 

Figure 1. Frequency response of the simply supported (0/45/-45/90)s rectangular plate 


2.5 
2 

1.5 
1 

0.5 
0 

-0.5 
-1 
-1.5 
-2 
-2.5 

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 

Po (psi) 

Figure 6. Plate centre response vs force amplitude at co = 1.45ci>i and T = 2.9 Ter 



1431 





1432 













1433 













1434 









1435 










1436 











GEOMETRICALLY NONLINEAR RESPONSE ANALYSIS 
OF LAMINATED COMPOSITE PLATES AND SHELLS 


C.W.S. Tot and B. Wangt 

Department of Mechanical Engineering 
University of Nebraska 
255 Walter Scott Engineering Center 
Lincoln, Nebraska 68588-0656 
U.S.A. 

E-mail; cwsto@unlinfo.unLedu 


Abstract 


The investigation reported in this presentation is concerned with the prediction 
of geometrically large nonlinear responses of laminated composite plate and 
shell structures under dynamic loads by employing the hybrid strain based flat 
triangular laminated composite shell finite elements. Large deformation of 
finite strain and finite rotation are emphasized. The finite element has eighteen 
degrees of freedom which encompass the important drilling degree of freedom 
at every node. It is hinged on the first order shear deformable lamination 
theory. Various typical laminated composite plate and shell structures under 
dynamic loads have been studied and representative ones are presented and 
discussed in this paper. Shear locking has not appeared and there is no zero 
energy mode detected in the problems studied. It is very accurate and efficient. 
Consequently, it is relatively much more attractive than other elements 
currently available in the literature for large scale nonlinear dynamic response 
analysis of laminated composite plate and shell structures. 


t Professor and corresponding author 
i Research Associate 


1437 


1. INTRODUCTION 


Many modem structures such as nuclear reactor containment installations, 
naval and aerospace structures, and their components, must be designed to 
withstand a variety of intensive dynamic disturbances. Because of their many 
attractive features over isotropic materials more and more stmctures or 
components in the aforementioned systems are made of laminated composite 
materials. The investigation reported in this paper is therefore concerned with 
the prediction of geometrically large nonlinear responses of laminated 
composite plate and shell structures, of complicated geometries, under transient 
excitations. With complicated geometries analytical solution is impossible and 
therefore a versatile numerical method, the finite element method has been 
employed. A hybrid strain based flat triangular laminated composite shell 
finite element has been developed by the authors [1,2] for the nonlinear 
analysis of plate and shell structures under static loadings. The present 
investigation is an extension of [1,2] to cases with the aforementioned 
dynamic forces. Among various attractive features of the derived element 
stiffness and consistent element mass matrices five are worthy of listing here 
for completeness. These are: (a) their ability to deal with large nonlinear 
elastic response of finite strain and finite rotation, (b) the fact that they are in 
explicit expressions and therefore no numerical integration is necessary, (c) the 
obtained results of a relatively comprehensive tests [2, 3] show that the 
element is free from shear locking, (d) the element gives correctly six rigid 
body modes, and (e) the finite element has three nodes and eighteen degrees 
of freedom (dof) which encompass the important drilling degree of freedom 
(ddof) at every node. It is based on the first order shear deformable lamination 
theory. It is a generalization of the low-order flat triangular shell element for 
isotropic materials developed earlier by Liu and To [4]. 

It is noted that one of the earlier work that employed triangular shell 
element is due to Noor and Mathers [5]. In the latter a mixed type triangular 
element was proposed. The element has six nodes, and 78 dof. It was based 
on the shallow shell theory and was shear deformable. Recently, Lin et al [6] 
developed a finite element procedure to analyze composite bridges. The finite 
element procedure was based on small elasto-plastic strains and updated 
Lagrangian formulation. The element used was flat and constructed by the 
superposition of a discrete Kirchhoff bending element and a linear strain 
triangular membrane element. It has six nodes. There are three translational 
and three rotational dof at its comer nodes and three translational dof at mid¬ 
side nodes. In 1994 a flat triangular shell element was presented for static 
nonlinear analysis by Madenci and Barut [7]. It is based on the so-called free 
formulation concept for analyzing geometrically nonlinear thin composite 
shells. A corotation form of the updated Lagrangian formulation is utilized. 
The theoretical basis was on the geometrically nonlinear Kirchhoff plate theory 
without considering the effects of transverse shear deformation. The element 


1438 


is of displacement type. It has three nodes and six dof for each node. While 
such formulation has some advantageous features computationally the element 
is relatively less efficient because (a) the linear element stiffness matrix 
consists of a basic and a higher-order stiffness matrices in the sense of 
Bergan and Nygard [8] for isotropic materials, and (b) the important effects 
of transverse shear deformation in the plate component of of this element has 
been disregarded. A more recent contribution on triangular elements is made 
by Zhu [9]. The natural approach is used to construct a curved triangular shell 
element for static analysis of geometrically nonlinear sandwich and composite 
shell structures. The element has six nodes. There are six dof at each corner 
node and three dof at each mid-side node. Updated Lagrangian description was 
adopted in the procedure. In the element formulation the transverse shear 
deformation was considered by assuming constant transverse shear stress 
distribution. 

In the next section the formulation of element stiffness matrices is 
outlined. Section 3 deals with the derivation of element matrices. Section 4 is 
concerned with the application of the derived elements to three example 
problems of plate and shell structures. The concluding remarks are included 
in Section 5. 


2. FORMULATION OF ELEMENT STIFFNESS MATRICES 

Finite element formulation for the derivation of a family of simple 
three-node, six dof per node, hybrid strain based laminated composite 
triangular shell finite elements for large scale geometrically nonlinear analysis 
is briefly outlined in this section. Large deflection of finite strains and finite 
rotations are included. The first order shear deformation theory and the 
degenerated three dimensional solid concept are adopted. In particular, element 
matrices for one member of the family are derived explicitly with the 
symbolic computer algebra package MACSYMA. To minimize the algebraic 
manipulation involved in the derivation, updated Lagrangian description is 
employed in the incremental formulation of the finite element procedure. In 
essence, the present formulation is an extension of the work by Liu and To [4] 
for isotropic materials to multi-layer laminated composite shells. Therefore, in 
the development the present approach follows closely that of the last reference. 

2.1 Incremental variational principle 

The Hellinger-Reissner functional tChr can be written as 



where, 

e' is the independently assumed strain field; 

e“ is the strain due to displacement; 

C is the material stiffness matrix or elasticity matrix; 

W is the work done by external forces, 
and the superscripts e and u indicate that the quantities are from independently 
assumed strain field and displacement field, respectively. For geometrically 
nonlinear analysis with incremental formulation and updated Lagrangian 
description, the static and kinematic variables in current equilibrium 
configuration at time t are assumed to be known quantities and the objective 
is to determine their values in the unknown subsequent equilibrium 
configuration at time t+At. For a time increment At, that is from time t to 
(t+At), one has 

- ATTjjj^CAUjAe®) - 7ifjjj(t+At) - ttHR® , 

or, with reference to equation (1), 

Atihr " / [(e')TC(Ae“) 

- i(Ae‘)fC(Ae') - 

where, 

Au is the vector of incremental displacement; 

Ae* is the vector of independently assumed incremental updated 
Green strains; 

Ae“ is the vector of incremental updated Green ’geometric’ strains 
or incremental Washizu strains; 

AW is the work-equivalent term corresponding to prescribed body 
forces and surface tractions in configuration C 

Equation (3) is the incremental form of Hellinger-Reissner variational 
principle. For updated Lagrangian description, the integral is evaluated at the 
current configuration C In the equation, the term 

J(Ae')^C(e^ - e“) dV 

Ve 


■f (Ae'')'^C(Ae“) 


(3) 


(Ae')'^C(e‘^ - e'^)] dV - AW . 


1440 


is the so-called compatibility-mismatch. Numerical results of Saleeb et al [10] 
showed that though totally discarding the term resulted in convergence 
difficulties, while including the term in only the first iteration of every load 
step yielded essentially the same results as those having the term under all 
circumstances. However, Liu and To [4] reported no difficulties for 
convergence when the term was ignored. In the current study, this term is also 
disregarded. Then equation (3) can be recasted as 


^^hr“/ [ + (Ae®)^C(Ae“) 

Vc 


(4) 


-■i(Ae')’'C(Ae')]dV-AW, 

2 


where = (e^)^C is the Cauchy (true) stress vector at the current 
configuration C ^ In this equation, the incremental Washizu strain Ae“ can be 
expressed in two parts 


Ae^ 


Aei" ^ ATii“ 


(5a) 


and they are related to the incremental displacement by 




|(Auy AUjj) . Arii” 


-AUyAU,^ 


(5b,c) 


where the Einstein summation convention for indices has been adopted and the 
differentiation is with respect to reference co-ordinates at the current 
configuration C 

Substituting equation (5a) into (4) yields 

A’'™ = f [- -(Ae')’'C(Ae') + (Ae')'^C(A£") + a'^Ae” 

V. 2 (6) 

+ o^Aii" + (Ae')fCATi“ ] dV - AW , 


where a is the Cauchy stress vector. 

Discarding the higher order term, ) ^Aq ^ results in 

AjIhr “ / [- -(Ae')'^C(Ae') + (Ae')TC(AE“) 

+ a^Ae'^ + a^Aq^] dV - AW . 


(7) 


1441 




2.2 Hybrid Strain Formulation 

Element stiffness matrices for a hybrid strain based finite element can 
be derived directly from equation (7). Generally the independently assumed 
strain field and displacement field can be written as 
Ae' = P Att , Au = <t) Aq 


where P is the strain distribution matrix, (|) is the displacement shape function 
matrix, Aa is the vector of incremental strain parameters and Aq is the 
incremental nodal displacement. Substituting equations (8a, b) into (7), and 
defining 

H - J P^CP dV^ , G, = / P'^CBl dV^ , 

V - / dVe . F. " / BlO dV. , 

V V, 


one can show that 

AT:jnj(Aq,Aa) = ’ ^Aa'^HAa + Aa'^G^Aq 

+ F^Aq + ■iAq^'k^LAq - F'^Aq ] 


( 10 ) 


where F is the external nodal force vector in the neighbour configuration 
associated with the AW term in equation (7); and B^l are the linear and 
nonlinear strain-displacement matrices, while Cq is the matrix containing the 
Cauchy stress components at the current configuration. 

Finally, one can show that 

(k[ . V) Aq - F{t. At) - F, , k[ = g7h-‘G. 0 


where the expression in equation (1 lb) is the element "linear" stiffness matrix. 
The term k^L defined in equation (9) is the "nonlinear" or initial stress 
stiffness matrix and Fi is the pseudo-force vector. The right hand side of 
equation (11a) is the equilibrium imbalance. 


1442 


3. ELEMENT MATRICES AND THEIR UPDATING 


The derivation of nonlinear element stiffness matrices, constitutive 
equations, mass matrices of the element shown in Figure 1 are outlined here. 
In addition, updating of configuration and stresses at every time step is 
considered here for completeness. 


3.1 Nonlinear Element Stiffness Matrices 

For the assumed displacement field, an arbitrary point within an 
element is governed by 


r*' 

3 

t 

h 



t 

Si 



0 


3 




( 12 ) 


The incremental displacements of an arbitrary point within the element are 


Au' 1 

1 O 

Au/| 

3 

< 

II 

AVi 1 

> + (AVi') . 

i-l 

Aw' , 

1 

Aw/ 

1*1 


Employing quadratic polynomials for the translational dof and including ddof 
lead to 


Au' ^ 

3 

AUi' 

3 

Ai(ii) Ai(i2) 


A0' 

Av' 

Aw' 

= E?i^ 

i-l 

AVi' 

Awj' 

^ c'E^i 

i-l 

Ai(21) Ai(22) 

Ai(31) Ai(32) ^ 


1-if 


0 

0 

Pi 


AO^i 

0 

0 

5i 

A 

Aei • 

. "Pi 

-5i 

0 


Ae' 


(14) 


1443 


In the foregoing, 

U = Ui^l + U2^2‘^U3^3 + Piejl + p20t2-"P3®G 

W = WiCl+W2^2-"W3^3-Pie,l-P2e ^- 536,3 

" 5 i®s 1 “^ 2 ®s 2 “^ 3 ®s 3 » 

0r = 0rl^l'*'®i2^2-*'®r3^3 
®s*®sl^l ■*'®s2^2'^®s3^3 
®tl^l '*’^ 12^2 ®e^3 


(15a) 


(15b) 


where 


Pi =■ (^1^3 "^12^2)^! ’ 5i “ (^31^3 “^12^2)^! » 

P2 * (^12^1 ”^3^3)^2 » ^2 * (^12^1 “^23^3)^2 ’ 

P3 “ (^3^2 ~ ^31 ^1)^3 » ^3 ” (^23^2 1)^3 • 

The remaining symbols have been defined by Liu and To [4] and are not 

repeated here for brevity. 

For the assumed strain field, the strain vector in equation (8a) may be 
written as 


Ae*^ 


Ae; 

4 

Ay' 




(16) 


where 

A^'m = PmA«„ . Ax' - PfcAa, 

Ay' - PsA«s 


(17a) 


1444 


with 


Aa^-lAttj Aa 2 Actj)’'’ , 

(17b) 


Aa^-{Aa4Aa5Aa^}'^,Aa^-{Aa7Aa8Aa,}’', 


and 



1 

0 

o' 


1 

0 

o' 

B 

II 

0 

1 

0 

11 

pf 

0 

1 

0 


o, 

0 

1. 


0 

0 

1 


- 83 ( 1 - 2 ^ 2 ) 83 ( 1 - 2 ^ 1 ) 0 

-13(1-2^2) (r3-r2)(l-2^i) 12(1-2^3) 


where the subscripts m, b and s denote the membrane, bending and transverse 
shear components of P in equation (8). 

By defining 

= / PjA'P^ da , = f P3"C;P„ da , 

a a 

- / Pjc^P. da , - / Ps^E'P, da , 

a a 

= / PjB'P^ da , = / P,"CJP, da , (18) 

a a 

H„b - / PmB'Pb da , P,^CX da . 

a a 

Hbb = / Pb'D'Pb da 


1445 


where a is the area of the triangular shell element and 


A' - E (C.)k(h,-h,.,). B' - i ^ (C,)k(h^h,l,), 

k-1 ^ k-1 

" 'T S (^a)k(^k“'^k-l) ’ (^b)k(^k“^k-l) ’ 

d k-1 k-1 

Ca - i: (c j,aik“Vi) > Cb - -i i: (c 

k-1 k-1 


(19) 


in which the integer n is the number of laminae in the laminated composite 
structure. Then the matrix H in equation (9) becomes 







H = 

Hta 


Hu 




H* 


9x9 


( 20 ) 


Similarly by defining 



/ pJa'B^ da . 

Gsm - / Ps"cX <ia 


a 

a 

Gms ■ 

/ da , 

= / Ps^E'B, da , 


a 

a 

Gbm = 

/ P,^B'B„ da , 

Gi, ■= / pXb, da . 


a 

a 

Gmb “ 

/ pJb'B, da , 

G., - / PsXX ^ . 


a a 


G,, = / pJd'B, da 


1446 


one has the matrix 



+ 

G„b 

+ 

G^ 


B 

X) 

o 

+ 

Gbb 

+ 

Gbs 


Gs. 

+ 

G.b 

+ 


9x18 


( 22 ) 


Therefore, with the ddof considered the element stiffness matrix can be shown 
to be 


k - k[ 


+ k^ + k^L 


(23) 


where the linear element stiffness matrix k^’ and the "nonlinear" or initial 
stress stiffness matrix k^L ^re defined by in equations (9) and (11), while the 
stiffness matrix associated with the ddof k^d is defined as 


kda = i:(G.),(h,A-.)/B7B,da, 


(24) 


k-1 


in which 

Bd = [Bdi Bj2 B^Ji^ig 


and 


Bdi 


-Ui.r 0 0 0 


with i = 1,2,3. 

The "nonlinear" or initial stress stiffness matrix k^L can be obtained if 
the nonlinear strain-displacement matrix B^l and the matrix Gc which contains 
the Cauchy stress components at the current configuration are available. The 
matrix B^l is defined by equation (45) of Liu and To [4]. 

The matrix Oq is constructed from the Cauchy stress vector c and 
defined as 


^11^ 


°3ll3 


^22^3 

” 23 '^ 


*^23^ 

O 3 


(25) 


1447 



with I 3 being the 3x3 identity matrix and O 3 a 3x3 null matrix. The transverse 
stress components of a are considered constant over the thickness, and all 
components of a are calculated and updated for each time step at the centroid 
of each element. 

3.2 Constitutive Equations 

For finite strain problems in the elastic range, the reduced stiffness 
matrix is a function of stresses. To incorporate finite strains in the analysis, 
several approaches can be applied. The following adopted from reference [4] 
is to add the linear elastic matrix a correction matrix which is a function of 
Cauchy stress. To begin with, the correction terms in tensor form becomes 

Ciid " - * OjiSfl + OnSik + "jiSik ) 


where 8 ^^ is the Kronecker delta. Note that this equation comes as a result of 
transforming the Jaumann stress rate to the incremental second Piola-Kirchhoff 
stress. If the stress and strain vectors are 

O = { Ojj O 22 O 33 0^2 ^23 Ojj } , 

e = { ®22 ®33 ®12 ®23 ®31 


the matrix form of equation (26) is 






0 

0 

2012 

0 

2013 


0 

4<J22 

0 

20i2 

20,3 

0 

1 

0 

0 

4033 

0 

2023 

2013 

2 

20i2 

2a,2 

0 

^ll'^^22 

Oi 3 

^23 


0 

2°23 

2023 

°13 

^22"^ ^33 

^12 


2oj3 

0 

20j3 

^^23 

^12 

Oii + O 


In present investigation the so-called degenerated concept is adopted and 
therefore the elastic modulus in the normal direction to the plane of the shell 
structures is considered zero. Consequently the stress and strain in the 
transversal direction are ignored. In the linear analysis the constitutive 
relations for a lamina have been defined as 

o = Qe (29) 


1448 


where 


°n ° zx ’ 

e - { e, By e„ )■" 


Qn 

Qi 2 

Qi 6 

0 

0 

Qi 2 

Q22 

Q26 

0 

0 

Qi 6 

Q26 


0 

0 

0 

0 

0 

Q44 


0 

0 

0 

^5 

Q55 


The corresponding matrix from equation (28) is 



0 

2<’.y 

0 

2°. 

0 

40 y 

2<^.y 

2o 

yz 

0 


2% 



°yy 

0 

2^. 

® zx 


°xy 


0 


^xy 



The material stiffness matrix for a lamina thus becomes 

C = Q + C 


(30) 


(31) 


(32) 


(33) 


where 


t^aJ3x2 
t^aj2x3 J2x2. 


in which C„ C^, and = ^ba are given in equation (19). 

With the consideration of large deformation and finite strain, the 
constitutive equations for a multilayered structure or laminate can be written 
as 


1449 


or simply 


(34) 


' N ' 


■ A' 

B' 

Ca 



^ M 



D' 

Cb 

- 

X 

Qs 



Ub 

e' 

8x8 

. y . 


^ ®N 

where N, M and Qj are the vectors of stress resultants corresponding to 
membrane, bending and transverse shear, respectively. The matrices A’, B’, 
D’, E’, and Cg have been defined in equation (19). 

3.3 Element Mass Matrices and Updating of Configurations and 
Stresses 

In the present study, with the updated Lagrangian description, the 
consistent mass matrix is formulated in the current configuration C'. The mass 
matrix is then updated at each time step. The assumptions are that the angular 
velocities and accelerations are small enough to be discarded. By following the 
procedures of Liu and To [4] the consistent element mass matrix can be 
obtained as 


in which mt„ and m„t are translational and rotational components of the 
consistent element mass matrix, respectively. Matrix m^ is the part associated 
with the ddof. When it is used for the incremental formulation with updated 
Lagrangian description, updating relevant quantities at each incremental step 
are required before evaluating the mass matrix. All these mass matrices are 
obtained explicitly with the symbolic computer algebra package MACSYMA. 

For each incremental step, the configuration and stresses have to be 
updated. Details of the steps can be found in the reference by Liu and To [4] 
and therefore are not included here. However, it may be appropriate to point 
out that the linear consistent element matrix for multi-layer composites has 
been employed by the authors [11] for vibration analysis of plates and shells. 


4. EXAMPLES OF LAMINATED COMPOSITE PLATE AND SHELL 

There are two main objectives in this section. First, accuracy of results 
obtained by the presently derived element matrices is studied. Second, the 
validity and conceptual adequacy of the formulation and assumptions made in 


the derivation of element matrices are assessed. For brevity, one multi-layer 
plate, one multi-layer shell structure, and a cantilever panel with free end step 
moment are included here. More example problems can be found in To and 
Wang [3], and Wang and To [12]. 

4.1 Multi-Layer Plate Under Uniformly Distributed Step Disturbance 

The square plate considered has two layers. Its geometrical dimensions 
are: side length a = 2.438 m and total thickness h = 0.00635 m. Each layer of 
the laminate has equal thickness. The plate stacking scheme^is cross-ply 
(0/90). The layer material properties are: ^ = 6.8974 x N/m^, Ej = 25 
Gi 2 = G ,3 = 0.5 E 2 , G 23 = 0.2 E 2 , V 12 = 0.25 and density p = 2498.61 kg/m . 

It is supported by hinges at its four edges. At these edges U or V (note 
henceforth upper case of deformation variable refers to global co-ordinate) 
parallel to the edges are not constrained. These boundary conditions are 
denoted as BCl in reference [13]. For the purpose of direct comparison with 
the results reported in the latter reference, one quarter of the plate is modeled 
by a 4 X 4 D mesh (see Figure 1 for the definition of D mesh). Thus, the 
boundary conditions applied are: V = 0 ^ = 0.0 at AB, U = W = 0,, = 0.0 at 
BC, V = W = 0y = 0.0 at CD and U = 0y = 0.0 at AD. In addition, all 0, are 
constrained. After application of the boundary conditions there are 158 
unknowns in this case. 

The uniformly distributed transversal step disturbance with intensity po 
= 490.5 N/m^ is applied to the plate. In the analysis, the option of inclusion 
of directors [3, 12] and small strain are selected. The time step size is At = 
0.001 seconds. The responses at the centroid obtained by using the HLCTS 
element are plotted in Figure 2. They are compared with those reported by 
Reddy [13] in which results were obtained with a nine-node rectangular 
isoparametric element. In the latter transverse shear was considered. Excellent 
agreement can be observed. Before leaving this subsection it may be 
appropriate to mention that the nonlinear element stiffness matrix presented 
in reference [13] is nonsymmetric while the one derived in the present 
investigation is symmetric. In fact, when the system is conservative the 
nonlinear element stiffness matrix can be shown to be symmetric. 

4.2 Spherical Shell Segment Under A Uniformly Distributed Step 

Disturbance 

The geometry of the spherical shell is shown in Figure 3 in which the 
shell is simply supported. The geometrical properties are: radius R = 10.0 m, 
the side length of the projected plane b = 0.9996 m and the total thickness h 
= 0.01 m. The spherical shell is considered having two equal thickness layers 
and they have the (- 45/45) lamination scheme. The pertinent material 
properties are: Ej — 2.5x10^' N/m^ E 2 = 1.0x10^° N/m^, Gi 2 = G 13 = 0.5x10 ^ 
N/m^, G 23 = 0.2x10*° N/m^, Poisson’s ratio V 12 = 0.25 and density p = 1.0x10 
kg/m^. For comparison to results available in the literature one quarter of the 


1451 


shell is modeled by the proposed hybrid strain based shell element (identified 
as HLCTS for brevity and convenience) with 4 x 4 D mesh. The boundary 
conditions applied to the finite element model are: V = = 0.0 at line 

AB,V = W = ©, = 0.0atBC,U = W = ©y = 0.0 at DC and U = ©y = ©, = 0.0 
at AD. The number of equations to be solved after the application of the 
boundary conditions is 189. A distributed step pressure is applied to its outer 
surface (pointing toward the outer surface). It has an intensity p = 2000.0 
N/ml The time step used were 0.03 s, 0.01 s and 0.005 s. As there was no 
significant difference and for efficient reason throughout the computation the 
time step of 0.03 s was adopted. The nonlinear transient response at the apex 
(central point A of the shell) is obtained and plotted in Figure 4. The problem 
has been solved by Wu et al. [14] who applied a curved high-order 
quadrilateral shell element. The latter has 48 dof and was developed based on 
the classical lamination theory. It is observed that there is a discrepancy of 
about 8 %, with respect to the HLCTS element results, for the amplitudes 
between the two set of results. However, they have the same vibration period. 
It is believed that the present results are more accurate as the element used in 
the present investigation is shear deformable. 

4.3 Cantilever Panel With Free End Step Moment 

To demonstrate the use of the proposed shell element for structures 
undergoing large rotation and large deformation a four layer cross-ply 
cantilever panel is considered here. More computed results for this case can 
be found in references [3] and [12]. It is symmetrically laminated with the 
stacking scheme (0/90/90/0). Its geometrical properties are: L = 1.2 m, b = 0.1 
m and h = 0.01 m. The material used for this cantilever is the high modulus 
graphite/epoxy composite. Its properties are: Ej = 2.0685x10" N/m^, E 2 = 
5.1713x10'' N/m^ G,-, = 3.1028x10^ N/m^ G,, = G 23 = 2.5856x10^ N/m^ p = 
1605 kg/m^ and Poisson’s ratio Vj 2 = 0.25. A step moment M about an axis 
parallel to the width of the panel is applied to the free end. The amplitude of 
this moment is = 1000.00 N-m. 

The panel is discretized by a 12 x 1 A mesh. At the fixed end, all dof 
are constrained. The finite element model has 144 unknowns. 

The time step At = 0.001 s is employed in the trapezoidal rule direct 
integration. The nonlinear transient responses at the end of the cantilever are 
solved by selecting the options of director included, small strain and constant 
thickness in the digital computer program developed. The computed end 
deflections are plotted in Figure 5. As noted in reference [3,12], the inclusion 
of directors in the formulation [15] is crucial as the directors are important 
parameters that constitute the so-called "exact geometry" for large rotation 
problems. 


1452 


5. CONCLUDING REMARKS 


The hybrid strain based laminated composite flat triangular shell 
(HLCTS) element for the static analysis of geometrically nonlinear laminated 
composite plates and shells has been further developed and employed to solve 
various dynamic problems. A relatively comprehensive study for various plate 
and shell structures idealized by this element has been performed and three 
representative examples are included to demonstrate its accuracy, efficiency 
and conceptual adequacy. It is concluded that the HLCTS element is attractive 
for large scale finite element analysis and modelling of shell structures 
undergoing geometrically large deformation of finite strain and finite rotations. 


ACKNOWLEDGMENT 

The first author gratefully acknowledges the financial support in the 
form of a research grant from the Natural Sciences and Engineering Research 
Council of Canada. The results reported above were obtained in the course of 
the research while the authors were at the University of Western Ontario. 


REFERENCES 

1. To, C.W.S. and Wang, B., Nonlinear theory and incremental 
formulation of hybrid strain based composite laminated shell finite elements. 
Proc. Second Int. Conf. on Composites Engineering, August 21-24, 1995, New 
Orleans, Louisiana, pp. 757-758. 

2. Wang, B. and To, C.W.S., Finite element analysis of geometrically 
nonlinear composite laminated plates and shells. Proc. Second Int. Conf. on 
Composites Eng., August 21-24, 1995, New Orleans, Louisiana, pp. 791-792. 

3. To, C.W.S. and Wang, Hybrid strain based geometrically nonlinear 
laminated composite triangular shell elements, Part II: Numerical studies. 
Comp, and Struct. (Submitted), 1996. 

4. Liu, M.L. and To, C.W.S., Hybrid strain based three node flat 
triangular shell elements. Part I: Nonlinear theory and incremental formulation. 
Comput. Struct., 1995, 54, 1031-1056. 

5. Noor, A.K. and Mathers, M.D., Nonlinear finite element analysis 
of laminated composite shells. In Computational Methods in Nonlinear 
Mechanics (Ed. by J.T. Oden, E.B. Becker, R.R. Craig, R.S. Dunham, C.P. 
Johnson and W.L. Oberkampf). Proc. Int. Conf on Comput. Methods in 
Nonlinear Mechanics, Austin, TX, 1974. 

6. Lin, J.J., Fafard, M., Beaulieu, D. and Massicotte, B., Nonlinear 
analysis of composite bridges by the finite element method. Comput. Struct., 
1991,40. 1151-1167. 


1453 


7. Madenci, E. and Bamt, A., A Free-formulation-based flat shell 
element for nonlinear analysis of thin composite structures. Int. J. Numer. 
Meth. Engng., 1994, 37, 3825-3842. 

8. Bergan, P.G., and Nygard, M.K., Nonlinear shell analysis using free 
formulation finite elements. In Finite Element Methods for Nonlinear 
Problems, (edited by Bergan, P.G., Bathe, K.J., and Wunderlich, W.) Springer- 
Verlag, 1986. 

9. Zhu, J., Application of natural approach to nonlinear analysis of 
sandwich and composite plates and shells. Comput. Meth. Appl Mech. Engng., 
1995, 120, 355-388. 

10. Saleeb, A.F., Chang, T.Y., Graf, W. and Yingyeunyong, S., A 
hybrid/mixed model for nonlinear shell analysis and its applications to large- 
rotation problems, Int. J. Num.. Meth. Engng., 1990 29, 407-446. 

11. To, C.W.S. and Wang, B., Hybrid strain-based three-node flat 
triangular laminated composite shell elements for vibration analysis. J. Sound 
and Vibration (submitted), 1996. 

12. To, C.W.S. and Wang, B., Transient response analysis of 
geometrically nonlinear laminated composite shell structures. Proc. of Design 
Eng. Conf. and Computers in Eng. Conf. (edited by McCarthy, J.M.), August 
18-22, 1996, Irvine, California, 96-DETC/CIE-1623. 

13. Reddy, J.N., Geometrically nonlinear transient analysis of 
laminated composite plates. A.I.A.A. J., 1987, 21, 621-629. 

14. Wu, C.Y., Yang, T.Y. and Saigal, S., Free and forced nonlinear 
dynamics of composite shell structures. J. Comp. Mat, 1987, 21, 898-909. 

15. To, C.W.S. and Wang, B., Hybrid strain based geometrically 
nonlinear laminated composite triangular shell elements, Part I: Theory and 
element matrices. Computers and Structures (submitted), 1996. 


A mesh 


D mesh 



Figure 1 Flat triangular laminated composite shell element 


1454 



Central deflection lY 



Figure 2 Response of a cross-ply plate 



Figure 3 Spherical shell segment under a uniformly distributed load 


1455 



Central deflection -W 



Time b 


Figure 4 Apex response with quarter shell considered 



- 0,2 0.0 0.2 0.4 0.6 0.0 1.0 1.2 1.4 

X (m) 

Figure 5 Evolution of cantilever panel with free end step moment 


1456 




ANALYTICAL METHODS III 


THE FREE, IN-PLANE VIBRATION OF CIRCULAR RINGS 
WITH SMALL THICKNESS VARIATIONS 

R S Hwang, C H J Fox and S McWilliam 
Department of Mechanical Engineering, University of Nottingham, 
University Park, Nottingham NG7 2RD, England 

Abstract 

Geometric imperfections which cause thickness variations will always 
exist in nominally circular rings and cylinders due to limitations in 
manufacturing processes. The effects of circumferential thickness variations on 
the natural frequencies of in-plane vibration are studied. The circumferential 
variations in the inner and outer surfaces are describedj in a very general wayj 
by means of Fourier series. Novozhilov thin-shell theory is used in conjunction 
with the Rayleigh-Ritz method to obtain the natural frequencies. Results are 
presented which show the effects of single-harmonic variations in the inner and 
outer surface profiles, taking account of the profile amplitude of, and the spatial 
phasing between, the inner and outer profiles. The frequency factors calculated 
from the numerical method are in good agreement with those obtained from the 
Finite Element method. 


1. Introduction 

The free vibrations of circular rings or shells had been studied by many 
authors for over a century. The early theoretical works are summarised by 
Love [I]. Most of these works are restricted to perfect rings or shells. 
However, in practice, geometric imperfections (thickness variations and 
departure from true circularity) are produced in the manufacturing process. 
These affect the natural frequencies and mode shapes. It is weE known that in any 
truly axisymmetric structure the vibration modes occur in degenerate pairs which 
have equal natural frequencies and mode shapes which are spatially orthogonal but 
of indeterminate circumferential location. The main effects of thickness variations 
are to split the previous equal natural frequencies and remove the positional 
indeterminacy [2]. Although these effects are often practically unimportant, there 
are some applications (especially inertial sensors based on vibration rings or 
cylinders [3]) where the small frequency splits and fixing of the modal positions is 
of primary practical significance. There is therefore a requirement to be able to 
predict in detail the effects on vibrational behaviour due to small departures from 
perfect circularity of the kind produced by manufacturing tolerances. 


1457 



The vibration of imperfect bells and rings were studied m the general 
way using group theory [4,5]. In reference [5] the selection rules for frequency 
splitting of thin circular rings were presented qualitatively. In reference[6] the 
frequency splitting behaviour of a thin circular ring was investigated both 
experimentally and analytically by first order perturbation theory. In reference 
[7], the classical frequency equations, which are generally used to predict the 
natural frequencies of a thin circular ring, were modified to describe an 
eccentric ring by using the perturbation method. In reference [8], Fourier series 
functions were used to represent the circumferential thickness variations of an 
eccentric cylinder. Love thin-sheU theory, which is only strictly suitable for a 
perfect ring or cylinder, was applied to investigate the free vibration of non- 
circular shells. 

In this paper, the free in-plane vibrations of thin rings of rectangular 
cross section with circumferential variations in thickness are studied. The 
circumferential variations in the inner and outer surfaces are described, in a very 
general way, by means of Fourier series. Novozhilov thin-shell theory [9], in 
conjunction with the well-known Rayleigh-Ritz method, are applied to analyse 
the vibration characteristics for in-plane flexural vibration of the ring which is 
considered as a special case of a thin shell [2,6]. The numerical method is used 
to investigate the effect of single-harmonic circumferential variations in the 
inner and outer surface profiles. The effects of harmonic number, amplitude and 
spatial phasing between the inner and outer profiles are investigated. Some 
important trends and patterns of effects of profile variations on the splitting of 
the natural frequencies are observed. The results obtained by using the 
numerical method developed in the current investigation are validated by 
comparison with Finite Element predictions. 


2. Method of Analysis 
2.1 Geometry 

Consider a thin ring of mean radius having a rectangular cross- 
section of mean thickness h (« r«) and axial length L (« Ta). The inner and 
outer surface vaiy along the global circumferential direction (Figure 1). rp 
denotes the distance from the centre of the mean radius of the ring to the point 
F on the middle surface. 

Two coordinate systems are used in the formulation of the equation of 
deformation. 


1458 


* Global polar coordinates (a', p', These are dir^ted along the global 
axial, circumferential and radial directions. The initial geometry of the 
undeformed, imperfect ring is defined using this coordinate system. 

* Local curvilinear coordinates (a, p, These are directed along the local 
axial, tangential and normal directions relative to the true middle surface and 
coincident with the principal coordinates of the middle surface. This local 
coordinate system is required for implementation of Novozhilov shell theory 
which specifies displacements in the local tangential and normal directions. 

is the angle between the global and local coordinate systems at the point P 
of the middle surface. 

All the displacements, thicknesses, and radii in this paper are expressed 
dimensionlessly by dividing by lo , where k is the representative length and is 
defmed as the mean radius of the ring. 



Figure 1. A thin ring having circumferentially arbitrary surfaces 

The shape of the middle surface of the ring is determined by the inner 
and outer surfaces which can be expressed by Fourier series as follows: 

f*(?')=U+'^f*cos(ip) + '^f; sin(jp) (1) 

i=I j=I 

/TP’)=/«'+ X/r«osc«P'-)+X/7««o‘P') (2) 

i=/ 


1459 






where / and / "(P') denote respectively the outer and inner surface 
functions with respect to the global circumferential coordinate P and fo ,// j 
//, fo', fi' and// are the Fourier coefficients which are defined in the usual 
way [10]. 

The middle surface of a shell or ring is defined as the locus of the points 
which lie at equal distances, * and from the outer and inner surfaces along 
the direction normal to the mid surface (see Figure 2). 


.. ^ 



Figure 2. The bounding surfaces and the middle surface 


For given inner and outer surfaces, / (PpO and/'^(p/»0 defined in the 
global coordinate system, the true middle surface can be determined using an 
iterative numerical procedure which is fully described in [11]. Once the point P 
on the true middle surface has been determined, the corresponding , Rp, 

Tp and Pp can be calculated- These will be used in the step-by step integrations 
which determine the strain energy and kinetic energy of the ring. 


2.2 Equations of Motion 


The strain energy for a thin ring whose length is much smaller than the 
mean radius takes the form [2,6]: 

S = «(!->■ l/R)d^d^ (3) 


1460 



Based on Novozhilov thin-shell theory, the normal strain epp in equation 
(3) is given as 


" 7 t / T> ^ 

(4) 

1 + ^/ R 


Ep= Vj^/R + w/R 

(5) 

Kp= - 1/R ( IV, p/j? +v/R), p 

(6) 


where Ep, Kp characterise the deformation of the middle surface of the thin ring 
and subscript “ , p “ denotes partial derivatives with respect to (5. Epis the strain 
tangential to the middle surface and Kp is the change of curvature, v, w are the 
nondimensional local displacement components of the point P on the middle 
surface along the tangential and normal directions respectively. 

Substituting equations (4)-(6) into equation (3), then integrating with 
respect to the thickness from hi~ to neglecting the 4th and higher powers 

of /rF^ and ^ f and noting that F(p) [- hi~^]d?t = 0 where Ffp) is 

an arbitrary function of p, the strain energy of a thin ring can be derived in 
terms of the local displacements v as follows: 

S= {[( V, p/ + 2 h'v, p + w^]( 1 /R)[hi* -hi~ ] 

2 rp; 

jK 

+ [ 2WW, p + 2w, p IV, pp - 2v w - 2viv, pp ] -h 

3K 

+ [(w,^)^ + v‘.2vw,f]^ ((l/R),f)^[kC^ -hr^]}d^ ( 7 ) 

jK 

Similarly, the kinetic energy of a thin ring, based on Novozhilov thin- 
shell shell theory, can be expressed as follows: 

T= {[(y,,)\(y^,,f]R[hC + 

2 '•Pi 

[3(v,t f- 4v,tW,,p + ( IV,/p f f (8) 

in which p is the density of the ring, and the subscript ‘,/ * denotes the partial 
derivatives with respect to time. 


1461 


For free vibration the tangential displacement v and the normal 
displacement w 'which satisfy the boundary condition can be assumed to take the 
following forms reg)ectively: 


V = sinnp-v^ cosnp)^^^^ (9) 

n 

w = cosnp + w^ sinnp)^^^^ (10) 

n 

where v„ and are the undetermined amplitude coefficients of the tangential 
and normal displacements of the middle surface respectively. The superscripts 
"s" and "c” refer to the fact that these coefficients are multiplied by sine and 
cosine terms respectively. 

Substituting equations (9) and (10) into equations (7) and (8), then 
applying the Rayleigh-Ritz procedure, the general frequency equation of the 
free ■vibrations for a thin ring is obtained and can be expressed in the following 
general matrix form: 



-A 

'M" 

Ui~ 


~0~ 






0 


where jfi: and M represent stiffiiess and mass matrices of size 2(N+1), and q 
denotes a vector of generalised coordinates v„, w„ etc. The matrix elements in 
equation (11) are given in [11]. Since in the general case the cross-section of 
the ring 'wdl not be symmetric with respect to P = 0, the classification of the 
modes as being “symmetric” and “antisymmetric” is meaningless. In the special 
case of a perfect circular ring, the off-diagonal terms [K^J, /M"/ and 
/Af7 appearing in equation (11) are null matrices, then equation (11) can be 
uncoupled into two equations: one is for the symmetric modes and the other is 
for the anti-symmetric modes -with respect to P=0. 


The frequency factors of the ring. An , are the eigenvalues of equation 
(11) and are defined by 


A 


0)1 d P 
E 


( 12 ) 


where a)„ is the natural frequency of the nth radial mode. The frequency factor 
An is proportional to the square of natural frequency . 


For a given value of n equation (11) will yield a pair of values of An. 
These will be equal in the case of a perfect ring but will be slightly different in 


1462 



the case of an imperfect ring, giving rise to a higher frequency mode and a 
lower frequency mode for each value of 

It should be noted that the matrix elements in equation (11) are 
expressed as integrals of the functions , Rp, yp and ^p with respect to 

the tangential coordinate These functions are expressed in terms of the local 
coordinates. Hence it is necessary to make a transformation to express these 
functions and integrals in terms of the global coordinates, so that the integrals 
can be evaluated over the global circumferential coordinate p* from 0 to 27t. 


3. Results and Discussion 

By using different combinations of trigonometric functions in equations 
(1) and (2), it is in principle possible to model any closed thin ring. For the 
purpose of illustration we will consider a nominally circular ring with a single 
harmonic variation in the inner and outer surfaces, given by 

(P') ~ 

/" OT = + V (JP’ - <!>) 

where h/ and hf are the ampHtudes of the unperfections of the outer surface 
and the inner surface measured from the mean outer radius ra and the mean 
inner radius r/ respectively, is the spatial phase angle between the 
trigonometric functions of the inner and outer surfaces at p* =0 , and /, j are the 
harmonic numbers of the surface variations. Figure 3 illustrates i =] = 3 for 
three values of ^ . 



(^ = 0 


<j> = 7t/2 


^ % 


Figure 3. Different spatial phase angles ^ for i = j = 3 


1463 





Results for the combinations of the geometric imperfections of i -J - 2, 
3, 4, 5, 6, hf* - hf = O.lhy OMlh , and <J» = 0,7c/4, nil, SjtM, tz are presented 
here. The ring dimensions and material properties are as follows: = 

40.75mm, r/ = 37.83mm, L = 2 ram = 206.7x10’ NW, p = 7850 kg/ml 

Note that hf = O.lh corresponds to a departure from circular which is 
much larger than would occur in practice due to imperfection. The results for hf — 
OJh are presented to highlight the effects. Practically however, hf = OJDlh 
represents a more realistic variation in thickness. 

Convergence studies indicated that for hf*, hf~ = O.lh the use of 30 terms 
in the solution series (equations 9 and 10) gave 4 significant figure accuracy or 
better for the frequency factors A* fox k - 0,1, ... 6. This was considered to be 
acceptable for the purposes of the illustrative examples considered here. 

In a parallel Finite Element study, beam elements, and two- and three- 
dimensional plane stress elements used to model an imperfect ring. In order to 
get 4 significant figures or better, 120 elements were used to model the 
complete ring. Comparison of the results obtained from the numerical method 
and the Finite Element Method shows that 

(i) there is good agreement between the curves of frequency factors 

obtained by the Finite Element method and the numerical method. 

(ii) the trends and patterns of frequency splitting are nearly identical 

irrespective of the analysis methods or the types of finite elements used. 

In considering the effect of single harmonic variations of the profile of the 
inner and outer surfaces on the natural fiequencies of different radial modes, the 
discussion will focus on three aspects: 

(a) , the effect of the harmonic number of the profile; 

(b) . the effect of the magnitude of the profile variations; and 

(c) . the effect of the spatial phasing between the profile variations of the 

inner and outer surfaces. 

The frequency splits shown are often very small (~ 0.001%). Note 
however that in some inertial sensor applications, such small frequency splits 
may be of practical significance. 


1464 



(a) The effect of profile harmonic number 

Table 1 compares the frequency factors A„ obtained for a perfect ring 
and an imperfect ring for ^-0, "^12, tc, h/" -hj — 0,lh, and i=j = 2 to 6. It is 
evident from these results that: 

For the flexural modes {n>2): 

(i) When i , j are equal and even (see Table 1), frequency splitting only 
occurs in the nth mode where w = Id. I 2 and /c is an integer. The 
maximum frequency splitting occurs in the n—H 2 modes (i.e. k ^ 1 ) 
and the splitting decays as k increases. 

(ii) When i , j are equal and odd (see Table 1) frequency splitting only 
occurs in the nth mode where n = ki and k ism integer. The maximum 
frequency splitting occurs for A: = 1, and splitting decreases as k 
increases. 

It should be noted here that frequency splitting in the higher modes 
exists but is very small, e.g., for <}» = 0, i - j = 2 and kf= OJh , frequency 
splitting occurs in the 2nd and higher radial modes. It can be seen from Table 1 
that the splits in frequency factor are 0.019% at the 2nd mode, 0.001% at the 
3rd mode, and less than 0.001% at the 4th mode or higher mode. These 
correspond to actual frequency splits of about 0.01%, 0.(XK)5% and less than 
0.CKX)5% respectively (equation (12)). 

For the radial extensional mode(« = 0), no frequency splitting occurs. 

It is clear from Table 1 that the trends and patterns of frequency splitting 
are the same for ^-0,n/2 and tc. However, frequency splitting is less foT^ = 0 
than for <}) = 71 under the same conditions. Frequency splits for ^ between 0 and 
K are intermediate between those for <{» =0 and <{) = 7t. 

The above patterns are in agreement with the qualitative results 
published in reference [5] in which only the conditions for non-splitting are 
established. 


1465 


Table 1 


The difference of frequency factors A on the radial 
modes w(n)[the parameters of profile variations are 
taken as hf=0.1h/ i=j=2 to $ and (a)«|>=0; (b)i^5=TC/2; (c)4>=?c] 


(a) 4>= 

=0 

A(0) 

A(2) 

A{3) 

A(4) 

A(5) 

A(6) 

Perfect 

1.0005 

0.003313 

0.02645 

0.09693 

0.2525 

0.5406 

i= 2 

high 

low 

-0.011% 

-0.023% 

-0.042% 

-0.041% 

-0.042% 

-0,038% 

-0.036% 

-0.035% 

i= 3 

high 

low 

0.075% 

-0.047% 

-0.088% 

-0.153% 

-0.133% 

-0.108% 

-0.094% 

-0.095% 

i= 4 

high 

low 

0.457% 

2.155% 

-2,287% 

-0.134% 

-0.150% 

-0.594% 

-0.349% 

-0.240% 

-0.248% 

i= 5 

high 

low 

1.631% 

-0.171% 

-0.141% 

-0.332% 

-0.229% 

-1.816% 

-0.838% 

i= 6 

high 

low 

5.656% 

-0.171% 

2.406% 

-2.726% 

-0.267% 

-0.715% 

-0.302% 

-5’.624% 


(b) 4)=Tr/2 

A(0) 

A(2) 

A{3) 

A(4) 

A{5) 

A(6) 

i= 

2 

high 

low 

0.007% 

0.336% 

-2.427% 

-0.595% 

-0.676% 

-0.971% 

-0.972% 

-1.169% 

-1.272% 

i= 

3 

high 

low 

0.090% 

-1.231% 

0.336% 

-2.559% 

-0.721% 

-0.869% 

-1.038% 

-1.040% 

i= 

4 

high 

low 

0.382% 

14.58% 

-18.71% 

-1.150% 

0.287% 

-2.871% 

-0.868% 

-0.828% 

-0.901% 

i = 

5 

high 

low 

1.236% 

-5.642% 

-1.239% 

-1.201% 

0.212% 

-3.703% 

-1.172% 

i = 

6 

high 

low 

4.111% 

-5.620% 

13.12% 

-17.59% 

-1.316% 

-1.404% 

0.117% 

-6.370% 


(c) 41=71 


i = 

2 

high 

low 

A{0) 

0.026% 

A(2) 

0.690% 

-4.827% 

A(3) 

-1.132% 

-1.340% 

A(4) 

-1,908% 

-1.912% 

A(5) 

-2.308% 

A{6) 

-2.514% 

i = 

3 

high 

low 

0.106% 

-2.469% 

0.709% 

-4.934% 

-1.312% 

-1.637% 

-1.989% 

-1.993% 

i= 

4 

high 

low 

0.314% 

18.81% 

-26.96% 

-2.227% 

0.528% 

-4.993% 

-1.388% 

-1.447% 

-1.532% 

i= 

5 

high 

low 

0.881% 

-10.89% 

-2.400% 

-2.135% 

-0,018% 

-5.031% 

-1.510% 

i= 

6 

high 

low 

2.808% 

-10.81% 

16.78% 

-25.36% 

-2.396% 

-2.134% 

-1.881% 

-5.060% 


Note: 


1. difference = [A (n) - A (n)porfc.ct]x 100% / A (n)perfect 

i\o 

2. A(n) = ——(H (n) , where (ii(n) is the natural frequency at 

E 

the nth radial mode. 


1466 



Table 2. The difference of frequency factors A on the radial 
inodes w(n) [the parameters of profile variations are 
taken as hf=0.01h, i=j=2 to 6 and (a)<|>=0; (b)<|)=TC/2; (c)<|)=7t ] 


(a) (|)=0 

Perfect 

A(0) 

1.0005 

A(2) 

0.003313 

A(3) 

0.02645 

A{4) 

0.09693 

A(5) 

0.2525 

A(6) 

0.5406 

i= 2 high 
low 

=0 

=0 

=0 

=0 

=0 

=0 

i= 3 high 
low 

0.001% 

«0 

-0.001% 

-0.002% 

-0.001% 

-0.001% 

-0.001% 

i= 4 high 
low 

0.005% 

0.222% 

-0.223% 

-0.001% 

-0.001% 

-0.006% 

-0.004% 

-0.002% 

i= 5 high 
low 

0.016% 

-0 . 002% 

-0.002% 

-0.003% 

-0.002% 
-0.019% 

-0.009% 

i= 6 high 
low 

0.060% 

-0.002% 

0.256% 

-0.260% 

-0.003% 

-0.008% 

-0.003% 
-0-. 063% 


(b) <j)=7t/2 

A{0) 

A(2) 

A(3) 

A(4) 

A(5) 

A(6) 

i= 2 

high 

low 

=0 

0.004% 

-0.024% 

-0.006% 

-0.010% 

-0.012% 

-0.013% 

i= 3 

high 

low 

0.001% 

-0.012% 

0.003% 

-0.026% 

-0.007% 

-0.009% 

-0.010% 

i= 4 

high 

low 

0.004% 

1.684% 

-1.725% 

-0.011% 

0.003% 
-0.029% 

-0.009% 

-0.009% 

i= 5 

high 

low 

0.012% 

-0.058% 

-0.012% 

-0.012% 

0.002% 

-0.038% 

-0.012% 

i= 6 

high 

low 

0.044% 

-0.057% 

1.550% 

-1.595% 

-0.013% 

-0.014% 

0.001% 

-0.069% 


(C) <j>=7t 


i= 2 

high 

low 

A{0) 

=0 

A(2) 

0.008% 
-0 . 048% 

A(3) 

-0.012% 

A(4) 

-0.019% 

A(5) 

-0.023% 

A(6) 

-0.025% 

i= 3 

high 

low 

0.001% 

-0 . 024% 

0.008% 

-0.050% 

-0.013% 

-0.016% 

-0.020% 

i= 4 

high 

low 

0.003% 

2.360% 

-2.440% 

-0.021% 

0.006% 
-0.050% 

-0.014% 

-0.015% 

i= 5 

high 

low 

0.009% 

-0.114% 

-0.023% 

-0.020% 

«0 

-0.051% 

-0.015% 

i= 6 

high 

low 

0.028% 

-0.112% 

2.165% 

-2.251% 

-0.023% 

-0.020% 

-0.018% 
-0.051% 


Note: 1. difference = [A (n) - A (n)p*r£ecc]x lOO^s / A(n}p<,r£«ot 

2. A(n) = (£t^(n). where tOfn) is the natural frequency of 

E 

the nth radial mode. 


1467 



(b) The elBfect of proBle amplitude 

The effects of varied profile amplitude (h/' = hf = O.lh and OMlh) upon 
the frequency factors for ^ = 0, it and i =J = 2 to 6 can be seen by 
comparing Tables 1 and 2. It may be concluded from these results and others 
which are presented in [11] that: 

(1) When n-i 12, frequency factor splitting due to variable profile magnitude 

compared with the frequency factor of the perfect ring is nearly 
proportional to the profile amplitudes, hj* and hj . 

For example, for i ^ = 7 t and hf^ = hf 0,lh ,0.01h (see Tables 1 

and 2), the magnitude of frequency splitting at the 2nd mode is 45.77% for 
hf = OJh and 4.80% for hf= OMlh. These correspond to actual frequency 
splits of 24% and 2.4% respectively (equation (12)) 

(2) For modes other than those for which n^i 12, splitting of frequency factors 

is nearly proportional to the square of the profile amplitudes, and hf . 

For example, for i “4, 2 nd hf* = hf — O.lh ,0Mlh (see Tables 1 
and 2), the magnitude of frequency factor splitting at the 4th mode are 
5.521% for hf-O.lh and 0.056% for A/= O.Olh. 

These results shown in Tables 1 and 2 for (j) = 0 and = ^t/2 show that 
the general nature of the trends regarding the effect of profile amplitude 
variations on the frequency factors are the same for all values of ^ , although 
the magnitudes of the changes in frequency factors depend on ({>, as discussed in 
the following section. 

(c) The effect of spatial phase angle variations 

The effects of the variations of spatial phase angle <j) on the frequency 
factors are shown in Figure 4, from which it is evident that 

(1) As frequency splitting occurs (see Figure 4.a-4.d ), the maximum frequency 
splitting is obtained at <) = tc and the minimum splitting occurs at (p = 0. It 
is clear that the maximum frequency splitting occurs in the n — il2 modes. 

(2) In modes for which no frequency splitting occurs (see Figure 4.e and 4.f), 

the minimum frequency difference compared with that of the perfect ring is 
detected at <j) = 0 and the maximum at ^ = n. Irrespective of the value of 
(j) , the frequencies of these modes are always less than the corresponding 
frequencies of the perfect ring. 


1468 



WKemnceIn 4 (%) at(eiwic»ln A (%) 



Figure 4. Effect of spatial phasing on frequency factors with hf = O.lh and 
(a) i=j=:4,2nd mode; (b) i=:j=6,3rd mode 
(c) i=j=2, 2nd mode ; (d) i=j=3,3rd mode 
(e) i=j=3, 4th mode ; (f) i=j=4, 5th mode 


1469 

















4, Conclusions 


In this paper, Novozhilov thin-shell theory and the Rayleigh-Ritz procedure 
have been applied to derive the frequency equations of a thin ring with a rectangular 
cross-section and a circumferential proffle variatioa Profile variations are 
represented, in the general way, by Fourier series functions and the method gives 
quantitative predictions of frequency splitting. The observed firequency splitting 
patterns are in agreement with previously published qualitative results. Numerical 
results have been presented for example cases in which the inner and outer profiles 
are nominally circular with superimposed single-harmonic variations m radius. The 
effects on frequency splitting of the harmonic number of the profile variation, and 
the amplitude and spatial phasing between the inner and outer surfaces have been 
investigated. 


References 

1. Love, A-E.H., A Treatise on the Mathematical Theory of Elasticity^ Dover 

Publications, New York, fourth edition, 1952. 

2. Fox, A simple theory for the analysis and correction of frequency 

splitting in slightly imperfection rings. Journal of Sound and Vibration, 
1990,142(2), 227-43. 

3- Fox, Vibrating cylinder rate gyro: theory of operation and error 

analysis. Proceeding of DGON Symposium on Gyro Technology, 
Stuttgart, 1988, Chapter 5. 

4. Chamley, T. and Perrin, R., Studies with an eccentric bell. Journal of Sound 

and Vibration, 1978, 58(4), 517-25. 

5. Perrin, R., Selection rules for the splitting of the degenerate pairs of natural 

frequencies of thin circular rings. Acustica, 1971, 25, 69-72. 

6. Chamley, T. and Perrin, R., Perturbation studies with a thin circular ring. 
Acustica, 1973, 28, 139-46. 

7. Valkering, T.P. and Chamley, T., Radial vibrations of eccentric rings. 
Journal of Sound and Vibration, 1983,86(3), 369-93. 

8. Tonin, R.F. and Bies, D.A., Free vibration of circular cylinders of variation 

thickness. Journal of Sound and Vibration, 1979, 62(2), 165-80. 

9. Novozhilov, V.V., The Theory of Thin Shells, P. Noordhoff Ltd., The 
Netherlands, 1959. 

10. Kreyszig, E., Advanced Engineering Mathematics, John Wiley & Sons, 
Inc., Singapore, 1993, pp. 569-71. 

11. Hwang, R., Free vibrations of a thin ring having circumferential profile 
variations. Ph.D. Thesis, University of Nottingham, U,K, (in preparation) 


1470 


FREE VIBRATION ANALYSIS OF TRANSVERSE-SHEAR 
DEFORMABLE RECTANGULAR PLATES 
RESTING ON UNIFORM LATERAL ELASTIC EDGE SUPPORT 

D.J. Gorman 

University of Ottawa 
770 King Edward Ave., 

Ottawa, Canada KIN 6N5 


ABSTRACT 

Utilizing the Superposition Method a free vibration analysis is conducted for 
transverse-shear deformable rectangular plates resting on uniformly distributed 
lateral elastic edge support. Edges are free of moment. The thick isotropic 
Mindlin plate is utilized for illustrative purposes. The Mindlin equations are 
satisfied throughout. Typical computed results are plotted for a square plate. 


INTRODUCTION 

It is well accepted that classical rectangular plate boundary conditions denoted 
as simply supported or clamped are often not achieved in real structures. This is 
because of elasticity in the edge supports. Furthermore, in many rectangular 
plate installations elastic edge supports may be utilized intentionally. For this 
reason a number of studies of effects of elasticity in the edge supports on 
rectangular plate free vibration frequencies have been conducted and results 
published. Almost all of these studies have been devoted to the free vibration 
behaviour of thin isotropic plates. Studies by the author, related to this family of 
vibration problems, have been devoted to situations where elastic stiffness is 
uniformly distributed along the edges as well as cases where the stiffnesses are 
arbitrarily distributed. All of his studies have been conducted by means or the 
Superposition Method and in a fairly recent article he has demonstrated that all 
of these families are amenable to analytical type solutions [1]. 

In this paper we exploit the powerful Superposition Method to analyse the free 
vibration behaviour of transverse-shear deformable plates resting on uniform 
lateral elastic edge support. This represents a much more complicated problem 
than the thin isotropic plate problems discussed above. For our purposes we 
choose the thick shear-deformable Mindlin plate and base our solution on 
Mindlin theory. 

In the interest of keeping the literature review up to date the recent publication 


1471 


of SAHA, KAR, and DATTA [2] is drawn to the attention of the reader. They 
report on a study of thick Mindlin plates resting on edge supports with uniform 
lateral and rotational elasticity. They have employed a Rayleigh-Rite energy 
approach. Plate lateral displacement is represented by a rather complicated set 
of Timoshenko beam functions, each extremity of each beam being attached to 
a local lateral and torsional spring. It will be seen that no such functions need be 
selected in the superposition approach adopted here. Another related paper is 
one by the present author dealing with Mindlin plates where lateral displacement 
along the plate edges is forbidden but uniform rotational elastic support is 
provided [3]. This problem is somewhat easier to solve since edge lateral 
displacement is forbidden and, unlike the present problem, mixed derivatives do 
not show up in the boundary condition formulation. This latter problem was 
shown to be amenable to solution by the modified Superposition-Galerkin 
Method which is extremely easy to use when it is applicable. 


MATHEMATICAL PROCEEDURE 

A solution to the present problem is obtained through th& supei-position of the 
eic^ht edge-driven forced vibration solutions (building blocks) shown 
scheraaticaly in Figure 1. All of the non-driven edges have slip-shear support. 
This type of support, indicated in the figure by two small circles adjacent to the 
edc^e, implies that the edge is free of torsional moment and transverse shear 
fomes. Furthermore, rotation of the plate cross-section along the edge is 
everywhere zero. 

We begin by examining the first building block. Its driven edge is free of 
torsional moment, and rotation of the plate cross-section along this boundary is 
every where zero. This latter condition is indicated by two solid dots adjacent to 
the edge. Driving of this edge is accomplished by a distributed harmonic 
transverse sheai* force of circular frequency o). The spatial distribution of the 
shear force is expressed as, 

Q,l,.,= E E„cos(m-l)7t5 ( 1 ) 

m = 1,2 


where k is the number of terms required in the series. 


1472 



Fig. 1 Schematic representation of building blocks utilised in theoretical 

analysis. 

We now examine the response of the above building block to this harmonic 
excitation. The proceedure followed is almost identical to that described in an 
earlier publication [4]. A concise description will be provided here for the sake 
of completeness. 

The governing differential equations which control the response of thick Mindlin 
plates are written in dimensionless form as, 

W d" W ^ ^ ^ W = 0 (2) 

8 ^“ (j)- art“ ({) ari V 3 


d^- 4)- 8 Ti^ 4) 5 ^ 8 r| i ^ 5 U 


(3) 


8f 4)^v, a-n- 4)Vi d^dr\ 



1473 


Transverse shear forces, bending moments, etc., are written as. 


V 9 „ 


^ 3^ cj) 3r| ^ 3 ti 


(j) 3n 

at a tf 9 <|i- 1 a i(r. 


When subjected to the first term of the driving force (Eqn 1) the response 
of the building block will be essentially that of a Timoshenko beam. The 
governing differential equations reduce to a set of two which may be 
written as 


d“W ^ d-ijj A'^(})“c|);;W 

-+ (p-!• +- 

d rt- d Ti"“ V3 


= 0 


(6) 


and 

d'l};^ V3<{)V ^ 1 dW K 

^2 dn j 12 


(7) 


It is convenient to represent the lateral displacement W, and plate cross- 
section rotation as, 

W(Ti) = X(Ti), and (ti) = Z (t)) 

The governing differential equations may then be written as 

X"(q) + a^,Z'(q) + b„,X(ti) = 0 (8) 


and 

Z''(q) + a,^3X'(n) + b„,5Z(Ti) = 0 (9) 

where superscripts imply differentiation with respect to r|. Coefficients 


1474 


.... etc,, are defined in reference [4]. 

Applying the appropriate differential operators to this set of equations 
the parameter X (ti) is eliminated and a second order ordinary 
homogenous differential equation is obtained involving the parameter Z 
(q). It is found that for our range of interest the roots of the characteristic 
equation associated with this differential equation are always real. There 
are then three possible pairs of roots depending on the coefficients in the 
above differential equation. Designating these pairs of real roots as 
Rj and possible forms of solution exist as follows, 

Casel, RpR2<0-0 Case2, R,<0-0; R,>0-0 CaseS, Ri,R2>0-0 (lO) 

In all work reported here it has been found that only, case 2, has been 
encountered. 

It will be obvious that the functions X (q) and Z (q) must be symmetric 
with respect to the ^ axis. We may therefore write for case 2, 

X(q) = cos aq + cosh pq, (11) 


and 

Z(^) = Am ( cos a q + cosh P q (12) 

where a = ^|R,| , and P = 

Expressions and S^.^re obtained by taking advantage of the 
coupling of equations 8 and 9, as was done in Reference [4]. 

We then impose the boundary conditions, Q^ = E,^^,and ilf^ = 0,at q=l, 
in order to evaluate the unknowns and B^^^ of Equation (11). 
Accordingly we obtain, 

E 

X(q) = .^{cos aq + X 1 cosh Pq}, (13) 


and 


1475 



Z(T 1 ) = ^ {S„, sin a n + XI S„, sinh P t|}, (14) 


where XI and X2 are easily evaluated. 

Next we examine the response of the first building block to driving terms 
where m>l. We follow the proceedure described in Reference [4]. 


Levy type solutions for the parameters W, .... 

. etc., are written as. 

W(5,ti) = XjTi) cosirni? 

(15) 

litres,Tl) = Y„(ri) sinm-ii^ 

• (16) 

= Z„(T|) cosmn? 

(17) 


It will be noted that all required boundary conditions along the edges, ^ 
= 0, and ^ = 1/ are satisfied. 


Next, the above expressions (Eqns 15,16,17) are substituted in the set of 
governing differential equations. The following set of coupled ordinary 
differential equations, written in matrix form are obtained 


X 


0 

0 



X./' 


t>n,l 

^m2 

0 


x„ 


o' 

Y 

^ m 


0 

0 

^m2 


Y.' 



^m4 

0 


Y„, 

►_ 4 

0 

zj 

. m J 


.^m3 

^m4 

0 




0 

0 

bn,5. 




0 


(18) 


Again, the quantities .... etc., are defined in Reference [4]. 

Applying the appropriate operators on the above equations, as was done 
in Reference [4] we are able isolate a single homogenous sixth order 
ordinary differential equation involving the dependent variable (ti), 
only. Because first, third, and fifth order derivates are missing from this 
equation the associated characteristic equation can be formulated as a 
cubic algebraic equation. Again it is found that for the range of the 


1476 



present study all of the roots are real Designating these roots^ as 

R ,R,, and R3, it follows that four solution cases are possible depending 

orl the coefficients of the characteristic equation. They are. 

Case!, R,, R,, and R. < 0-0 Case2, R,, R^ < 0-0 ; R3 > 0-0 

‘ - (19) 

Cases, Rj < 0-0 ; R, and R 3 > 0-0 Case4, Rp R, and R 3 > 0-0 


Inthepresentstudyonlycase3,andcase4are€ncountered.Introdudng a =^\Rj 1 , 

P = , and y = JIR 31 and recognizing that (q)must be 

antisyrnmetric about the ^ axis while X^(q)and Y^(q)must be 
symmetric, we are able to write for case 4, 


Y^(q) = A,^ cosh aq+B^^ cosh Pq + C^^^ cosh yq 


( 20 ) 


Utilizing the coupling of the ordinary differential equations, as in 
Reference [4], it follows that we may write, 

X^/q) = A^R,^i cosh ocq + B^R^, N + (21) 


and 

Z^(q) = A^S^, cosh ocq + B^^S^, cosh Pq+C,^iS ^3 cosh yq (22) 

The quantities R^,,, S^p ... etc., are evaluated following steps described 
in Reference [4]. Expressions for (q), X^^ (q), etc., for case 3 will differ 
from the above expressions only iii that Cosh a q must be replaced by cos 
a q. 

The unknown constants A^, B „p etc., of the above solutions are evaluated 
by enforcement of boundary conditions along the edge, q=l. These 
conditions comprise zero torsional moment, zero edge rotation, with 
transverse shear force = E^p For case 4 we obtain 

(n) = — {cosh a q + X1 cosh p q + X 2 cosh y q} (23) 


with the functions X^(q) and (q) differing from Y,^(q) only in that 


1477 


R^j and S^j, etc., must be included. 

We therefore now have the exact response of the first building block to 
the imposed driving force components available. It will be observed in 
Figure 1 that the second, fifth, and sixth building blocks differ from the 
first only in that they are driven along different edges. Solutions for their 
response are therefore easily extracted from that of the first. 

Focusing our attention on the third building block we find that its driven 
edge is free of transverse shear forces and torsional moment. It is driven 
by a distributed cross-section harmonic rotation. The spacial distribution 
of this imposed driving rotation is also represented by the series of 
Equation 1. 

The reader will appreciate that a solution for the third building block is 
obtained by following steps identical to those described for the first. Only 
the imposed boimdary conditions along the driven edge differ. Solutions 
for the quantities W, etc., will be identical in form to those already 
developed for the first building block except that quantities XI, XI, etc., 
will be slightly different. We designate them as XIP, X2P, etc., for the 
edge-rotation driven building blocks. Solutions for the remaining four 
building blocks of Figure 1 are therefore available. 


THE EIGENVALUE MATRIX 

This matrix is shown schematicaly in Figure 2. It is generated following 
established practices. Let us first consider the transverse force 
equilibrium condition along the edge, rj = 1. It is readily shown that this 
equilibrium condition is written in dimensionless form as, 

Q, + K„W = 0 (24) 


The plus sign of this equation must be replaced by a minus sign when we 
formulate the corresponding equations for the edges, 11 = 0 , and ^=0. 


1478 



Em 

1 2 3 

^ En 

1 2 3 

c:> 

CM 

C'> - 

W 4 

^ Ep 

i 2 3 

- 1 10 

^ E, 

1 2 3 

r&ean-, 
% < 

^ Es 

1 2 3 

1 2 3 


- 



- 

- - ■ 

- 

::: 

- 

: : : 

LvJ 

: : : 

- 

- - - 

- 

_ - - 

- 

: : : 

- 



- 

« - - 

- 

Ill 

- 

: : : 

- 

1 

1 M 1 

t 

p- 

- ^ - 


- - - 

- 

: : : 


: : : 

- 

□ 

- 

: : : 

- 


- 


- 

: : : 

*_ 

[v 

+ 

r 

_ - - 

- 

- - - 

- 

- “ " 

- 

: : : 

■ 

- 

: : : 

- 

: : : 


Ill 

- 

: : : 

1 M 

1 

i" 

j 

]' 

: : : 

- 

: : : 

- 

" " I 

■ 

I I - 






Pig 2 Schematic representation of Eigenvalue matrix based on 

three-term function expansions. Short bars indicate non¬ 
zero elements. M or V on inserts to right indicate edges 
along which moment or lateral force equilibrium is 
enforced. 

To construct the first three equations upon which this matrix is based we 
superimpose all eight building blocks and expand their net contribution 
to displacement W in a cosine series. The transverse shear force along the 
edge, 11=1, is already available in such a series. We then express the left 
hand side of equation 24 in series form and require that each net 
coefficient in this series must vanish. This leads to 3 homogenous 
algebraic equations relating the 8 k imknowns where, for the illustrative 
matrix of Figure 2, k equals 3. 

A second set of three homogenous algebraic equations is obtained by 
enforcing the corresponding lateral equilibrium condition along the edge. 


1479 


^ = 1. Moving down the matrix of Figure 2 it is seen that a third and 
fourth set of equations are obtained by enforcing the moment 
equilibrium condition, i.e., net bending moment equals zero along the 
same edges, in an identical fashion. Finally, it is seen in Figure 2 that four 
more sets of equations are obtained by enforcing the required 
equilibrium conditions along the edges, ti=0, and ^=0. 

We thus have, in general, 8 k homogenous algebraic equations relating 
the 8 k unknown driving coefficients. The coefficient matrix of this total 
set of equations forms our Eigenvalue matrix. 

Certain measures can be taken to greatly simplify and expedite 
generation of the matrix. It will be observed (Fig. 2) that the matrix is 
composed of 64 natural segments. This array of segments may be 
referred to through the indices (I, J). It is expedient to first generate the 
matrix without including contributions related to the driving shear 
forces along the building block edges (Eqn. 24). The matrix is then 
completed by adding the quantity 1.0 to diagonal elements of segments 
(1,1) and (2,2), and subtracting 1.0 from the diagonal elements of 
segments (5,5) and (6,6). 

Physical reasoning leads also to another vast signification. One may 
begin by generating the elements of the matrix lying below the first four 
building blocks, only, (Fig. 2). Following a proceedure as discussed in 
Reference [4], and exercising caution with respect to necessary sign 
changes, all of the remaining segments of the matrix may be extracted 
from those already generated. 

Eigenvalues are, of course, those values of the dimensionless frequency, 
X-, which cause the determinant of the Eigenvalue matrix to vanish. 
Mode shapes are obtained after setting one of the non-zero driving 
coefficients equal to zero and solving for the others. 


PRESENTATION OF COMPUTED RESULTS 

It will be appreciated that problems involving vast arrays of stiffness 
coefficients, plate aspect ratios, thickness-to-length ratios, etc. can be 
resolved by the proceedure described above. Only a single typical 
problem and its solution will be discussed here, for illustrative purposes. 

We consider a square plate with equal dimensionless lateral elastic 


1480 


stiffness imposed along each edge. Results are presented for two 
thickness-to-length ratios, 0.01, and 0.1. Two important observations 
may be made before examining these results. First, for the very thin plate 
of thickness-to-length ratio, 0.01, we expect the Eigenvalue vs edge- 
stiffness ratio curves to almost co-inside with those for a thin isotropic 
plate based on thin plate theory. Secondly, we recall that the Eigenvalue 
limits for a thin plate will equal 0.0, and 2 as the elastic stiffnesse 
approaches natural limits of 0.0, and infinity. 



Results of a free vibration study of the above plate are presented in 
Figure 3. It will be noted that computed Eigenvalues are plotted against 
the parameter Kli / (j)J. By presenting data in this manner it is found 


1481 




that Eigenvalues for both the thick and the thin plate can be plotted on 
the same Figure. The absissa of the figure appears in five logarithmic 
decades. This range has been selected with a view to providing 
information for the